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+{"text":"\\section{Introduction and main results}\nLet $W_{t}$ be a Brownian motion in $\\mathbb{R}^{d}$ $(d\\geq1)$ with generator $\\Delta$\nand $T_{t}$ be an independent $\\alpha\/2$-stable subordinator\nwith $\\alpha\\in(0,2)$. Then the subordinate process\n$X_{t}:=W_{T_{t}}$ is an isotropic $\\alpha$-stable\nprocess and its infinitesimal generator is the fractional Laplacian operator $-(-\\Delta^{\\alpha\/2})$ which is given by\n$$\n-(-\\Delta^{\\alpha\/2})f(x):= \\int_{{\\mathbb R}^d}\\big[f(x+z)-f(x)-1_{|z|\\leqslant\n1}z\\cdot\\nabla f(x)\\big]\\frac{c_{d,\\alpha}}{|z|^{d+\\alpha}}{\\mathord{{\\rm d}}} z,\\quad\\forall f\\in C^2_c({\\mathbb R}^d),\n$$\nwhere $c_{d,\\alpha}$ is a positive constant.\nIt is well known that the heat kernel $p(t,x,y)$ of $-(-\\Delta^{\\alpha\/2})$ (which is\nalso the transition density of $X:=(X_t)_{t\\geqslant 0}$)\nhas the following estimates: for every $t>0$ and $x,y\\in{\\mathbb R}^d$,\n\\begin{align}\np(t,x,y)\\asymp\\left(t^{-d\/\\alpha}\\wedge\\frac{t}{|x-y|^{d+\\alpha}}\\right).  \\label{Heat}\n\\end{align}\nHere and below, for two non-negative functions $f$ and $g$, the notation $f \\asymp g$ means that there are positive constants $c_1$ and $c_2$ such that $c_1g(x)\\leqslant f(x)\\leqslant c_2g(x)$ in the common domain of $f$ and $g$.\n\n\nIn \\cite{Bo-Ja0}, by using Duhamel's formula,\nBogdan and Jakubowski studied the following perturbation of\n$-(-\\Delta^{\\alpha\/2})$ by a gradient operator:\n$$\n{\\mathscr L}^b:=-(-\\Delta^{\\alpha\/2})+b(x)\\cdot\\nabla,\\quad\\alpha\\in(1,2),\n$$\nwhere $b=(b^1, \\cdots, b^d): {\\mathbb R}^d\\to{\\mathbb R}^d$ with $b^j$, $j=1, \\dots, d$, belonging\nto the Kato class ${\\bf K}^{\\alpha-1}_d$ defined\nas follows: for $\\gamma>0$,\n$$\n{\\bf K}^{\\gamma}_d:=\\left\\{f\\in L^1_{loc}({\\mathbb R}^d): \\lim_{r\\downarrow 0}\\sup_{x\\in{\\mathbb R}^d}\n\\int_{B(x,r)}\\frac{|f(y)|}{|x-y|^{d-\\gamma}}{\\mathord{{\\rm d}}} y=0\\right\\},\n$$\nand $B(x,r)$ denotes the open ball centered at $x\\in\\mathbb{R}^{d}$ with radius $r$.\nLet $p^b(t, x, y)$ be the heat kernel of ${\\mathscr L}^b$.\nSmall time sharp two-sided estimates for $p^b$,  of the form \\eqref{Heat},\nwere established in \\cite[Theorems 1 and 2]{Bo-Ja0}.\nThe key points of the perturbation method used in \\cite{Bo-Ja0} are that, on one hand, a nice bound on $\\nabla_x p(t,x,y)$ is known, and on the other hand, the following 3-P inequality concerning $p(t,x,y)$ holds: there exists $C_0>0$ such that for any $0<s<t$ and $x,y,z\\in {\\mathbb R}^d$,\n\\begin{align}\n\\frac{p(t-s,x,z)p(s,z,y)}{p(t,x,y)}\\leqslant C_0\\Big(p(t-s,x,z)+p(s,z,y)\\Big).  \\label{ppp}\n\\end{align}\nSee also \\cite{C-K-K,Ch-Ku2,C-K-W,J-S,K-S,W-Z,Xi-Zh} and the references therein for two-sided heat kernel estimates of more general non-local operators in the whole space ${\\mathbb R}^d$.\n\n\n\\vspace{2mm}\nLet $D$ be an open subset of ${\\mathbb R}^d$, we can kill the process $X$ upon exiting D and obtain a\nsubprocess $X^D$ known as the killed isotropic $\\alpha$-stable process.\nThe infinitesimal generator of $X^D$ is the Dirichlet fractional Laplacian $-(-\\Delta)^{\\alpha\/2}|_D$,\nthat is, the fractional Laplacian with zero exterior condition.\nDue to the complication near the boundary, two-sided\nestimates for the Dirichlet heat kernel of $-(-\\Delta)^{\\alpha\/2}|_D$ (or equivalently, the transition density of $X^D$) are much more difficult to obtain. To state the related results, we first recall that an open set $D$ in ${\\mathbb R}^{d}$ is said to be $C^{1,1}$ if there exist $r_{0}>0$ and\n$\\Lambda>0$ such that for every $Q\\in\\partial D$, there exist a $C^{1,1}$-function $\\phi=\\phi_{Q}: \\mathbb{R}^{d-1}\\rightarrow\\mathbb{R}$ satisfying\n$\\phi(0)=\\nabla\\phi(0)=0$, $\\|\\nabla\\phi\\|_{\\infty}\\leqslant\\Lambda$, $|\\nabla\\phi(x)-\\nabla\\phi(z)|\\leqslant\\Lambda|x-z|$ and an orthonormal coordinate system $y= (y_{1}, \\cdots, y_{d-1},\ny_{d}):=(\\tilde{y}, y_{d})$ such that $B(Q, r_{0})\\cap D=B(Q,r_{0})\\cap \\{y: y_{d}>\\phi(\\tilde{y})\\}$. The pair $(r_{0},\\Lambda)$ is called the characteristics of the $C^{1,1}$ open set $D$. In \\cite{C-K-S-2}, Chen, Kim and Song proved that when $D$ is a $C^{1,1}$ open set in ${\\mathbb R}^d$, the heat kernel $p^D(t,x,y)$ of $-(-\\Delta)^{\\alpha\/2}|_D$ has the following two-sided\nestimates: for every $T>0$ and $(t,x,y)\\in(0,T]\\times D\\times D$,\n\\begin{align}\np^D(t,x,y)\\asymp\n\\left(1\\wedge\\frac{\\rho(x)^{\\alpha\/2}}{\\sqrt{t}}\\right)\\left(1\\wedge\\frac{\\rho(y)^{\\alpha\/2}}{\\sqrt{t}}\\right)p(t,x,y),\\label{Heat2}\n\\end{align}\nwhere $\\rho(x)$ denotes the distance between $x$ and $D^{c}$.\n\nBy reversing the order of subordination and killing, one can obtain a process $Y^D$ which is different from\n$X^D$. More precisely, we first kill the Brownian motion $W$ at $\\tau_{D}$, the first exit time of $W$ from $D$, and then subordinate the killed Brownian motion $W^D$ using the independent $\\alpha\/2$-stable subordinator $T_{t}$. That is, $Y^D:=(W^{D})_{T_{t}}$ is defined as\n\\begin{align*}\nY^D_{t}:=\n\\begin{cases}\nW_{T_{t}}, & T_{t}<\\tau_{D}\\\\\n\\partial, & T_{t}\\geqslant\\tau_{D}\n\\end{cases}\n=\n\\begin{cases}\nW_{T_{t}}, & t<A_{\\tau_{D}}\\\\\n\\partial, & t\\geqslant A_{\\tau_{D}},\n\\end{cases}\n\\end{align*}\nwhere $\\partial$ is a cemetery state, $A_{t}:=\\inf\\{s>0: T_{s}\\geqslant t\\}$ is the inverse of $T$ and the last equality follows from the fact $\\{T_{t}<\\tau_{D}\\}=\\{t<A_{\\tau_{D}}\\}$.\nThe process $Y^D$ is called a subordinate killed Brownian motion. For the differences and relationship between\nthe processes $X^{D}$ and $Y^D$, see \\cite{S-V2}. The infinitesimal generator of $Y^D$ is the spectral fractional Laplacian $-(-\\Delta|_{D})^{\\alpha\/2}$, which is defined as a fractional power of the negative Dirichlet Laplacian. It is a very useful object in analysis and partial differential equations (see \\cite{B-S-V,P-A,S-Z}) and has been intensively studied (see \\cite{A-D,D-M-Z,G-P-R-S-S-V,S-V} and the references therein).\nWhen $D$ is a bounded $C^{1,1}$ domain,\nthe following sharp estimates for the heat kernel $r^D(t,x,y)$ of $-(-\\Delta|_{D})^{\\alpha\/2}$ (which is also the transition density of $Y^D$) were obtained in \\cite[Theorem 4.7]{Song}: for every $T>0$ and $(t,x,y)\\in(0,T]\\times D\\times D$,\n\\begin{align*}\nr^D(t,x,y)\\asymp\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{\\left(|x-y|+t^{1\/\\alpha}\\right)^{2}}\\right)p(t,x,y).\n\\end{align*}\nIn Lemma \\ref{sharp} below, we will give the following\nalternative form of the estimates above: for $(t,x,y)\\in(0,T]\\times D\\times D$,\n\\begin{align*}\nr^D(t,x,y)\\asymp\\left(1\\wedge\\frac{\\rho(x)}{|x-y|+t^{1\/\\alpha}}\\right)\\left(1\\wedge\\frac{\\rho(y)}{|x-y|+t^{1\/\\alpha}}\\right)p(t,x,y),\n\\end{align*}\nwhich is more convenient to use.\n\n\n\\vspace{2mm}\nGradient perturbations of Dirichlet operators have also been widely studied in recent years.\nIn \\cite{C-K-S-1}, Chen, Kim and Song studied the following gradient perturbation of the Dirichlet fractional Laplacian:\n$$\n{\\mathscr L}^{b,D}:=\\left(-(-\\Delta)^{\\alpha\/2}+b(x)\\cdot\\nabla\\right)|_{D},\\quad\\alpha\\in(1,2).\n$$\nUnder the condition that $b\\in{\\bf K}^{\\alpha-1}_d$ and $D$ is a\nbounded $C^{1,1}$ open set in $\\mathbb{R}^{d}$ with $d\\geq2$, Chen,\nKim and Song \\cite[Theorem 1.3]{C-K-S-1} showed that the heat kernel $p^{b,D}(t,x,y)$ of ${\\mathscr L}^{b,D}$ has the same\nestimates as in (\\ref{Heat2}). This result was generalized to unbounded $C^{1,1}$ open sets by \\cite{P-R}. Unlike the whole space case, there was no good estimate on\n$\\nabla_x p^D(t, x, y)$, thus \\cite{C-K-S-1, P-R} used Duhamel's formula\nfor the Green function and the probabilistic road-map designed in \\cite{C-K-S-2} for establishing\nthe estimates \\eqref{Heat2}.\n\nIn the recent paper \\cite{K-R},  Kulczycki and Ryznar  proved the following gradient estimate\nfor $p^D(t, x, y)$: for any $T>0$, there exists a constant $C=C(d,T)>0$ such that for any $(t, x, y)\\in (0, T]\\times D\\times D$,\n\\begin{align*}\n|\\nabla_x p^{D}(t,x,y)|\\leqslant  \\frac{C}{\\rho(x)\\wedge\nt^{1\/\\alpha}}p^{D}(t,x,y).\n\\end{align*}\nUsing this result, we gave, in the recent preprint \\cite{C-S-X-X}, a direct proof of the main results in \\cite{C-K-S-1, P-R} by using Duhamel's formula, with drift $b=(b^1, \\cdots, b^d): D\\to {\\mathbb R}^d$, where each $b^j$, $j=1, \\dots, d$, belongs to the following  Kato class:\n$$\n{\\bf K}_{D}^{\\alpha-1}:=\\left\\{f\\in L^1_{loc}(D): \\lim_{r\\downarrow0}\\sup_{x\\in D}\\int_{D\\cap\nB(x,r)}\\frac{|f(y)|}{|x-y|^{d+1-\\alpha}} {\\mathord{{\\rm d}}} y=0\\right\\}.\n$$\nMoreover, we also obtain a gradient estimate for $p^{b,D}(t,x,y)$. Notice that by H\\\"older's inequality, $L^p(D)\\subseteq {\\bf K}_{D}^{\\alpha-1}$ provided $d\/(\\alpha-1)<p\\leqslant \\infty$.\n\n\\vspace{2mm}\nThe aim of this paper is to study the following perturbation of spectral fractional Laplacian by a time-dependent gradient operator:\n$$\n{\\mathscr L}^{D,b}:=-(-\\Delta|_D)^\\frac{\\alpha}{2}+b(t,x)\\cdot\\nabla,\\quad\\alpha\\in(1,2),\n$$\nwith $b(t, x)=(b^1(t, x), \\cdots, b^d(t, x)): (0, \\infty)\\times D\\to{\\mathbb R}^d$ satisfying certain conditions.\nWe shall derive sharp two-sided estimates for the heat kernel $r^{D, b}(t, x, y)$\nof ${\\mathscr L}^{D,b}$ in bounded $C^{1,1}$ domains.\nMoreover, we also obtain a\ngradient estimate as well as the H\\\"older continuity of the gradient of $r^{D, b}(t, x, y)$, which are of independent interest.\n\nTo state our main result, let us first introduce our local Kato\nclass of space-time functions used in this paper.\n\n\\begin{definition}\\label{Definition}\nLet $D$ be a domain in ${\\mathbb R}^{d}$\nand $\\gamma\\geqslant 0$. For a real-valued function $f$ on $(0,\n\\infty)\\times D$, we define for every $\\delta>0$,\n\\begin{align*}\nK^{\\gamma}_{f}(\\delta):=\\sup_{t>0,x\\in\nD}\\delta^{\\gamma\/\\alpha}\\!\\int_{0}^{\\delta}\\!\\!\\!\\int_{D}&\\big[s^{-\\gamma\/\\alpha}+(\\delta-s)^{-\\gamma\/\\alpha}\\big]\\left(1\\wedge\\frac{\\rho(y)}{|x-y|+s^{1\/\\alpha}}\\right)\\\\\n&\\quad\\times\\frac{s}{(|x-y|+s^{1\/\\alpha})^{d+\\alpha+1}}\\cdot|f(t\\pm s,y)|{\\mathord{{\\rm d}}} y{\\mathord{{\\rm d}}} s.\n\\end{align*}\nWe say that the function $f$ belongs to thel Kato class ${\\mathbb K}_{D}^{\\gamma}$ if\n$\\lim_{\\delta\\downarrow0}K_{f}^{\\gamma}(\\delta)=0$.\n\\end{definition}\n\n\\begin{remark}\nWe note that our Kato class is time-dependent, which is needed when consider parabolic problems, see \\cite{J-S,ZhQ}. One can easily check that if $0\\leqslant\\gamma_1<\\gamma_2$, then ${\\mathbb K}_D^{\\gamma_2}\\subseteq{\\mathbb K}_D^{\\gamma_1}$.\nBy Lemma \\ref{include} below, we have that ${\\bf K}_D^{\\alpha-1}\\subset {\\mathbb K}_D^{0}$\nand that, for  $1<p,q\\leqslant \\infty$, $L^q({\\mathbb R};L^p(D))\\subseteq {\\mathbb K}_D^{\\gamma}$ provided $\\tfrac{d}{\\alpha p}+\\tfrac{1}{q}<1-\\tfrac{1+\\gamma}{\\alpha}$.\n\\end{remark}\n\nIn the remainder of this paper, we always assume that $b=(b^1, \\cdots, b^d): (0, \\infty)\\times D\\to{\\mathbb R}^d$ and each $b^j$, $j=1, \\dots, d$, belongs to ${\\mathbb K}_{D}^0$ .\n\nBy Duhamel's formula, the heat\nkernel $r^{D,b}(s,x;t,y)$ of ${\\mathscr L}^{D,b}$ should satisfy the following integral equation: for $0\\leqslant s<t$ and $x,y\\in D$,\n\\begin{align}\\label{Duhamel}\nr^{D,b}(s,x;t,y)=r^D(t-s,x,y)+\\int_s^t\\!\\!\\!\\int_{D}r^{D,b}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr^D(t-r,z,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r,\n\\end{align}\nor\n\\begin{align}\\label{duhamel}\nr^{D,b}(s,x;t,y)=r^{D}(t-s,x,y)+\\int_s^t\\!\\!\\!\\int_D\nr^{D}(r-s,x,z)b(r,z)\\cdot\\nabla_z r^{D,b}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n\\end{align}\nNotice that in (\\ref{Duhamel}) the derivative of the unknown heat kernel is not involved, and hence easier to solve.\nWhile (\\ref{duhamel}) is connected directly to the mild solutions of the corresponding parabolic equations, and from which one can easily derive the H\\\"older continuity of the gradient of the unknown heat kernel.\nFor convenience, we define for $t>0$ and $x,y\\in D$,\n\\begin{equation}\\label{qq}\nq^D(t, x, y):=\\left(1\\wedge\\frac{\\rho(x)}{|x-y|+t^{1\/\\alpha}}\\right)\\left(1\\wedge\\frac{\\rho(y)}{|x-y|+t^{1\/\\alpha}}\\right)p(t,x,y).\n\\end{equation}\nThe following is the main result of this paper.\n\n\\begin{theorem}\\label{main}\nLet $D$ be a bounded $C^{1,1}$ domain in $\\mathbb{R}^{d}$\nand $b\\in{\\mathbb K}^{0}_{D}$.\nThen there exists a unique function $r^{D,b}(s,x;t,y)$ on $(0,\n\\infty)\\times D\\times D$ satisfying (\\ref{Duhamel}) such that:\n\n\\begin{enumerate}[(i)]\n\n\\item (Two-sided estimates)\nfor any $\\delta>0$,\nthere exists a constant $C_1>1$ such that for all $0\\leqslant s<t\\leqslant s+\\delta$\nand $x, y\\in D$, we have\n\\begin{align}\\label{estimate}\nC_1^{-1}q^D(t-s,x,y)\\leqslant r^{D,b}(s,x;t,y)\\leqslant C_1 q^D(t-s,x,y);\n\\end{align}\n\n\\item (Gradient estimate)\nfor any $\\delta>0$, there exists a constant $C_2>0$ such that for all $0\\leqslant s<t\\leqslant s+\\delta$\nand $x, y\\in D$,\n\\begin{align}\\label{vi}\n|\\nabla_x r^{D,b}(s,x;t,y)|\\leqslant C_2\n\\frac{1}{\\rho(x)\\wedge\\left(|x-y|+(t-s)^{1\/\\alpha}\\right)}q^D(t-s,x,y),\n\\end{align}\nand $r^{D,b}(s,x;t,y)$ also satisfies (\\ref{duhamel});\n\n\\item (C-K equation) for all $0\\leqslant s<r<t$ and $x,y\\in D$, the following Chapman-Kolmogorov's equation holds:\n\\begin{align}\\label{eq21}\n\\int_{D}r^{D,b}(s,x;r,z)r^{D,b}(r,z;t,y){\\mathord{{\\rm d}}} z=r^{D,b}(s,x;t,y);\n\\end{align}\n\n\\item (Generator) for any $f\\in C^2_c(D)$, we have\n\\begin{align}\\label{eq23}\nR_{s,t}^{D,b}f(x)=f(x)+\\int_s^tR_{s,r}^{D,b}\\mathcal{L}^{D,b}f(x){\\mathord{{\\rm d}}} r,\n\\end{align}\nwhere $R^{D,b}_{s,t}f(x):=\\int_{D}r^{D,b}(s,x;t,y)f(y){\\mathord{{\\rm d}}} y$;\n\n\\item (Continuity) for any uniformly continuous function $f(x)$ with compact supports, we have\n\\begin{align}\n\\lim_{t\\downarrow s} \\|R^{D,b}_{s,t}f-f\\|_\\infty=0;   \\label{con}\n\\end{align}\n\n\\item (H\\\"older continuity)\nif we further assume that $b\\in {\\mathbb K}_D^{\\gamma}$ for some $\\gamma\\in (0, \\alpha-1)$,\nthen for any $\\delta>0$,\nthere exists a constant $C_3>0$ such that for any $0\\leqslant s<t\\leqslant s+\\delta$\nand $x,x',y\\in D$, we have\n\\begin{align}\n|\\nabla_x r^{D,b}& (s,x;t,y)-\\nabla_x r^{D,b}(s,x';t,y)| \\leqslant C_3|x-x'|^\\gamma (t-s)^{-\\gamma\/\\alpha}\\nonumber\\\\\n&\\times\\frac{1}{\\rho(\\widetilde x)\\wedge\\left(|\\widetilde x-y|+(t-s)^{1\/\\alpha}\\right)}q^D(t-s,\\widetilde x,y),  \\label{ho}\n\\end{align}\nwhere $\\widetilde x$ stands the point among $x$ and $x'$ which is closer to $y$.\n\\end{enumerate}\n\\end{theorem}\n\nWe remark that the gradient estimates (\\ref{vi}) and (\\ref{ho}) are new even in the case $b\\equiv0$.\nWe now briefly describe the main idea of our argument.\nDue to the difference between the processes $Y^D$ and $X^D$,\nthe method used in \\cite{C-K-S-1,P-R}  does not work for ${\\mathscr L}^{D,b}$.\nInstead, we will use Duhamel's formula (\\ref{Duhamel}) to obtain the sharp two-sided estimates of the heat kernel. As mentioned before, two main ingredients are needed: the gradient estimate for $r^D(t,x,y)$ and the corresponding 3-P inequality, both of which are unknown. In fact, by Remark \\ref{pp} below, we shall see that the 3-P inequality of the form (\\ref{ppp}) does not hold for the heat kernel $r^D(t,x,y)$. Because of these, we will first derive an estimate on $\\nabla_x r^D(t,x,y)$, and then establish a generalized 3-P type inequality for $r^D(t,x,y)$.\nThe gradient estimate and the H\\\"older estimate for $r^{D,b}(s,x;t,y)$ follow as easy by-products of our perturbation argument.\n\n\\vspace{2mm}\nThe remainder of this paper is organized as follows. In Section 2, we\nprepare some important inequalities for $r^D(t,x,y)$ and derive its first and second order gradient estimates. The proof of the\nmain result, Theorem \\ref{main}, will be given in Section 3.\n\n\nWe conclude this introduction by spelling out some conventions that\nwill be used throughout this paper. The letter C with or without\nsubscripts will denote an unimportant constant and $f\\preceq g$\nmeans that $f\\leqslant Cg$ for some $C\\geq1$. The letter $\\mathbb{N}$\nwill denote the collection of positive integers, and\n$\\mathbb{N}_{0}=\\mathbb{N}\\cup\\{0\\}$. We will use $:=$ to denote a\ndefinition, and we assume that all the functions\nconsidered in this paper are Borel measurable.\n\n\n\\section{Estimates for $r^{D}(t,x,y)$}\n\n\nIn the remainder of this paper, $D$ denotes a bounded $C^{1,1}$ domain in ${\\mathbb R}^d$.\nFor simplicity, we first introduce some functions for latter\nuse. Given $d\\geqslant 1$, $\\vartheta\\in{\\mathbb R}$ and $\\alpha\\in(0,2]$, we define for $t>0$ and $x,y\\in D$,\n$$\n\\varrho_{d}^\\vartheta(t,x):=\\frac{t^\\vartheta}{(|x|+t^{1\/\\alpha})^{d+\\alpha}},\n$$\nand\n\\begin{align}\n\\hat q_\\alpha(t,x,y):=1\\wedge\\frac{\\rho(x)}{|x-y|+t^{1\/\\alpha}},\\quad q_\\alpha(t,x,y):=\\hat q_\\alpha(t,x,y)\\hat q_\\alpha(t,y,x).\\label{q}\n\\end{align}\nThen we have $p(t, x, y)\\asymp \\varrho_{d}^1(t,x-y)$ and $q^D(t, x, y)=q_\\alpha(t,x,y)\np(t, x, y)$.\n\nWe will first establish a generalized 3-P type inequality for $r^D(t,x,y)$, and then derive its first and second order\ngradient estimates, which will be essential in\nconstructing the solution to the integral equation (\\ref{Duhamel}).\n\n\\subsection{Generalized 3-P inequality}\nLet $T>0$ be fixed. Recall that $r^D(t,x,y)$ is the heat kernel of\n$-(-\\Delta|_D)^{\\frac{\\alpha}{2}}$, and for any $t\\in(0, T]$ and $x, y\\in\nD$, we have\n\\begin{align*}\nr^D(t,x,y)\\asymp\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{(|x-y|+t^{1\/\\alpha})^{2}}\\right)\\varrho^1_{d}(t,x-y).\n\\end{align*}\nThe estimates above are not very convenient for our application since $\\rho(x)$ and $\\rho(y)$\nare intertwined together.\nWe prove the following result.\n\n\\begin{lemma}\\label{sharp}\nFor any $t\\in(0,\nT]$ and $x, y\\in D$, we have\n\\begin{align}\nr^D(t,x,y)\\asymp q_\\alpha(t,x,y)\\varrho^1_{d}(t,x-y)\\asymp q^D(t, x, y).\n\\label{er}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nThe second comparison follows immediately the sentence after \\eqref{q}. So we will only prove\nthe first comparison.\nIt is obvious that\n$$\nq_\\alpha(t,x,y)\\preceq 1\\wedge\\frac{\\rho(x)\\rho(y)}{(|x-y|+t^{1\/\\alpha})^{2}}.\n$$\nThus we only need to show that\n\\begin{align}\nq_\\alpha(t,x,y)\\succeq 1\\wedge\\frac{\\rho(x)\\rho(y)}{(|x-y|+t^{1\/\\alpha})^{2}}.    \\label{00}\n\\end{align}\nOne can easily see that the above inequality holds when\n$$\n\\rho(x)\\vee\\rho(y)\\leqslant |x-y|+t^{1\/\\alpha}\\quad\\text{or}\\quad\\rho(x)\\wedge\\rho(y)\\geqslant|x-y|+t^{1\/\\alpha}.\n$$\nBy symmetry, it suffices to prove (\\ref{00}) in the case when\n$$\n\\rho(x)\\leqslant |x-y|+t^{1\/\\alpha}\\leqslant\\rho(y).\n$$\nUsing the fact that $\\rho(y)\\leqslant\\rho(x)+|x-y|$, we can deduce\n\\begin{align*}\n1\\wedge\\frac{\\rho(x)\\rho(y)}{(|x-y|+t^{1\/\\alpha})^{2}}&\\leqslant 1\\wedge\\frac{\\rho(x)(\\rho(x)+|x-y|)}{(|x-y|+t^{1\/\\alpha})^{2}}\\\\\n&\\leqslant 1\\wedge\\frac{\\rho(x)^2}{(|x-y|+t^{1\/\\alpha})^{2}}+1\\wedge\\frac{\\rho(x)\\cdot|x-y|}{(|x-y|+t^{1\/\\alpha})^{2}}\\\\\n&\\preceq 1\\wedge\\frac{\\rho(x)}{|x-y|+t^{1\/\\alpha}},\n\\end{align*}\nwhich implies the desired result.\n\\end{proof}\n\n\\begin{remark}\\label{pp}\nBy (\\ref{er}) and the same argument as in \\cite[Remark 2.3]{C-K-S-3}, one can see that for all $t\/4<s<3t\/4$ and $x,y,z\\in D$ with $2|x-y|\\geqslant |x-z|+|z-y|$, it holds that\n\\begin{align*}\n\\frac{r^D(t+s,x,y)[r^D(t,x,z)+r^D(s,z,y)]}{r^D(t,x,z)r^D(s,z,y)}&\\preceq \\left(\\frac{\\rho(x)[\\rho(z)+|x-y|+(t+s)^{1\/\\alpha}]}{\\rho(z)[\\rho(x)+|x-y|+(t+s)^{1\/\\alpha}]}\\right)\\\\\n&\\quad+\\left(\\frac{\\rho(y)[\\rho(z)+|x-y|+(t+s)^{1\/\\alpha}]}{\\rho(z)[\\rho(y)+|x-y|+(t+s)^{1\/\\alpha}]}\\right),\n\\end{align*}\nwhich goes to zero as $\\rho(x)=\\rho(y)\\rightarrow0$. This means that, unlike (\\ref{ppp}), the inequality\n$$\n\\frac{r^D(t,x,z)r^D(s,z,y)}{r^D(t+s,x,y)}\\preceq r^D(t,x,z)+r^D(s,z,y)\n$$\ncan not be true for all $t,s>0$ and $x,y,z\\in D$, even for balls.\n\\end{remark}\n\nWe now proceed to prove a generalized 3-P type inequality for $r^D(t,x,y)$. Let us start with the following result.\n\\begin{lemma} For any $t,s\\geqslant 0$ and\n$x,y,z\\in D$, we have\n\\begin{align}\n\\frac{q_\\alpha(t,x,z)q_\\alpha(s,z,y)}{q_\\alpha(t+s,x,y)}\\preceq [\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2. \\label{3q}\n\\end{align}\n\n\\end{lemma}\n\\begin{proof}\nNote that, for any $a, b>0$, it holds that\n\\begin{equation}\\label{elmentary}\n1\\wedge \\frac{a}{b}\\asymp \\frac{a}{a+b}.\n\\end{equation}\nThus,\n\\begin{align*}\n\\frac{\\hat q_\\alpha(t,x,z)\\hat q_\\alpha(s,y,z)}{q_\\alpha(t+s,x,y)}&\\asymp \\frac{\\left((t+s)^{1\/\\alpha}+|x-y|+\\rho(x)\\right)\n\\left((t+s)^{1\/\\alpha}+|x-y|+\\rho(y)\\right)}{\\left(t^{1\/\\alpha}+|x-z|+\\rho(x)\\right) \\left(s^{1\/\\alpha}+|z-y|+\\rho(y)\\right)}\\\\\n&\\preceq 1+\\frac{t^{1\/\\alpha}+|x-z|}{s^{1\/\\alpha}+|z-y|+\\rho(y)}+\\frac{s^{1\/\\alpha}+|z-y|}{t^{1\/\\alpha}+|x-z|+\\rho(x)}.\n\\end{align*}\nBy  (\\ref{q}), we have\n\\begin{align*}\n{\\mathcal I}:=\\frac{q_\\alpha(t,x,z)q_\\alpha(s,z,y)}{q_\\alpha(t+s,x,y)}&=\\frac{\\hat q_\\alpha(t,x,z)\\hat q_\\alpha(s,y,z)}{q_\\alpha(t+s,x,y)}\\hat q_\\alpha(t,z,x)\\hat q_\\alpha(s,z,y)\\\\\n&\\preceq \\hat q_\\alpha(t,z,x)\\hat q_\\alpha(s,z,y)+\\frac{\\rho(z)}{s^{1\/\\alpha}+|z-y|+\\rho(y)}\\hat q_\\alpha(s,z,y)\\\\\n&\\qquad+\\frac{\\rho(z)}{t^{1\/\\alpha}+|x-z|+\\rho(x)}\\hat q_\\alpha(t,z,x).\n\\end{align*}\nUsing the fact\n$$\n\\rho(x)+|x-z|\\asymp\\rho(z)+|x-z|,\n$$\nwe further get that\n$$\n\\frac{\\rho(z)}{t^{1\/\\alpha}+|x-z|+\\rho(x)}\\asymp\\frac{\\rho(z)}{t^{1\/\\alpha}+|x-z|+\\rho(z)}\\asymp \\hat q_\\alpha(t,z,x),\n$$\nand similarly\n$$\n\\frac{\\rho(z)}{s^{1\/\\alpha}+|z-y|+\\rho(y)}\\asymp\\frac{\\rho(z)}{s^{1\/\\alpha}+|z-y|+\\rho(z)}\\asymp \\hat q_\\alpha(s,z,y).\n$$\nThus, we have\n\\begin{align*}\n{\\mathcal I}\\preceq \\hat q_\\alpha(t,z,x)\\hat q_\\alpha(s,z,y)+[\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2\\preceq [\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2.\n\\end{align*}\nThe proof is finished.\n\\end{proof}\n\nAs a direct consequence, we can obtain the following generalized 3-P type inequality for $r^D(t,x,y)$.\n\n\\begin{lemma}\nLet $T>0$. For any $0\\leqslant s,t\\leqslant T $ and $x, y, z\\in D$, it holds that\n\\begin{align}\n\\frac{r^D(t,x,z)r^D(s,z,y)}{r^D(t+s,x,y)}\\preceq (t\\wedge s)\\Big(&[\\hat q_\\alpha(t,z,x)]^2\\varrho_{d}^0(t,x-z)+[\\hat q_\\alpha(s,z,y)]^2\\varrho_{d}^0(s,z-y)\\Big).   \\label{3r}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nCombining (\\ref{er}) and (\\ref{3q}), we get that\n\\begin{align*}\n{\\mathcal J}:=\\frac{r^D(t,x,z)r^D(s,z,y)}{r^D(t+s,x,y)}\\preceq \\Big([\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2\\Big)\n\\frac{\\varrho_{d}^1(t,x-z)\\varrho_{d}^1(s,z-y)}{\\varrho_{d}^1(t+s,x-y)}.\n\\end{align*}\nNote that\n\\begin{align*}\n\\big(|x-y|+(t+s)^{1\/\\alpha}\\big)^{d+\\alpha}\\preceq \\big(|x-z|+t^{1\/\\alpha}\\big)^{d+\\alpha}+\\big(|z-y|+s^{1\/\\alpha}\\big)^{d+\\alpha}.\n\\end{align*}\nThus\n\\begin{align}\n\\frac{\\varrho_{d}^1(t,x-z)\\varrho_{d}^1(s,z-y)}{\\varrho_{d}^1(t+s,x-y)}&=\\frac{t\\cdot s}{t+s}\\cdot\\frac{\\varrho_{d}^0(t,x-z)\\varrho_{d}^0(s,z-y)}{\\varrho_{d}^0(t+s,x-y)}\\nonumber\\\\\n&\\preceq (t\\wedge\ns)\\Big(\\varrho_{d}^0(t,x-z)+\\varrho_{d}^0(s,z-y)\\Big).  \\label{333}\n\\end{align}\nHence\n\\begin{align*}\n{\\mathcal J}&\\preceq (t\\wedge s)\\Big([\\hat q_\\alpha(t,z,x)]^2+[\\hat q_\\alpha(s,z,y)]^2\\Big)\\Big(\\varrho_{d}^0(t,x-z)+\\varrho_{d}^0(s,z-y)\\Big)\\\\\n&\\preceq (t\\wedge s)\\Big([\\hat q_\\alpha(t,z,x)]^2\\varrho_{d}^0(t,x-z)+[\\hat q_\\alpha(s,z,y)]^2\\varrho_{d}^0(s,z-y)\\Big),\n\\end{align*}\nwhere in the last inequality we have used the fact\n\\begin{align}\n\\hat q_\\alpha(t,z,x)\\preceq \\hat q_\\alpha(s,z,y)\\,\\,\\Leftrightarrow\\,\\,\\varrho_{d}^0(t,x-z)\\preceq \\varrho_{d}^0(s,z-y) \\label{semi}\n\\end{align}\nand the symmetry in $x$ and $y$. The proof is finished.\n\\end{proof}\n\n\\subsection{Gradient estimates}\nIn this subsection, we derive gradient estimates for $r^D(t,x,y)$.\nRecall that $r^D(t,x,y)$ is the transition density of $Y^D$. By the construction of $Y^D$,\nit holds (see \\cite[(2.2)]{Song}) that\n\\begin{align}\nr^D(t,x,y)=\\int_0^\\infty p^D_{2}(s,x,y)\\mu(t,s){\\mathord{{\\rm d}}} s, \\label{rtxy}\n\\end{align}\nwhere $p^{D}_{2}(t,x,y)$ is the Dirichlet heat kernel of $\\Delta|_{D}$, and $\\mu(t,s)$ is the density of the subordinator $T_t$. To derive gradient estimates for $r^D(t,x,y)$, we need to recall some estimates for $p^D_2(t,x,y)$.\n\nFor any $\\gamma, \\lambda\\in {\\mathbb R}$ and $(t, x)\\in (0, \\infty)\\times{\\mathbb R}^d$, we define\n$$\n\\xi^\\gamma_{\\lambda}(t,x):=t^{-(d+\\gamma)\/2}{\\mathrm{e}}^{-\\lambda |x|^2\/t}.\n$$\nIt is known (see \\cite[Theorems 3.1 and 3.2]{Song} for instance) there exist constants $\\lambda_1,\\lambda_2>0$, $C_1>1$ and $C_2<1$ such that\n\\begin{align}\np^{D}_{2}(t,x,y)&\\leqslant C_1\n\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{t}\\right)\\xi^0_{\\lambda_1}(t,x-y), \\quad (t, x, y)\\in (0, \\infty)\\times D\\times D,\\label{pd1ub}\\\\\np^{D}_{2}(t,x,y)&\\geqslant C_2\n\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{t}\\right)\\xi^0_{\\lambda_2}(t,x-y), \\quad  (t, x, y)\\in (0, 1]\\times D\\times D.\\label{pd1lb}\n\\end{align}\nMoreover, it follows from \\cite[Theorem 2.1]{ZQS} that, for any $T>0$,\nthere exists a constant $C_T>0$ such that for all $t\\in(0,T]$ and $x,y\\in D$,\n\\begin{equation}\n|\\nabla_x p^D_2(t,x,y)|\\leqslant \\left\\{\\begin{aligned}\n&\\frac{C_T}{\\rho(x)}p^D_2(t,x,y),\\qquad\\qquad\\qquad\\quad\\quad \\textrm{if}\\,\\,\\,\\rho(x)\\leqslant \\sqrt{t};\\\\\n&\\frac{C_T}{\\sqrt{t}}\\left(1+\\frac{|x-y|}{\\sqrt{t}}\\right)p^D_2(t,x,y),\\qquad \\textrm{if}\\,\\,\\,\\rho(x)>\\sqrt{t}.\\label{pd3}\n\\end{aligned}\n\\right.\n\\end{equation}\nIt turns out that  (\\ref{pd1ub}), \\eqref{pd1lb} and (\\ref{pd3}) are not very convenient to use.\nTo get easy-to-use forms of the estimates above, we first do some manipulations on $p^D_2(t,x,y)$.\nWe want to separate the terms $\\rho(x)$ and $\\rho(y)$.\nThe following elementary observation will be important.\n\n\\begin{lemma}\nFor any $\\lambda_2>\\lambda_1>0$ and $\\gamma\\in{\\mathbb R}$, it holds for all $t>0$ and $x,y\\in D$ that\n\\begin{align}\n\\left(1\\wedge\\frac{\\rho(x)\\rho(y)}{t}\\right)\\xi^\\gamma_{\\lambda_2}(t,x-y)\\preceq \\left(1\\wedge\\frac{\\rho(x)}{\\sqrt{t}}\\right)\n\\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^\\gamma_{\\lambda_1}(t,x-y).  \\label{pdd}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nIn light of \\eqref{elmentary}, it suffices to show that  for any $\\lambda_0>0$\n\\begin{align*}\n\\Big(\\rho(x)+\\sqrt{t}\\Big)\\left(\\rho(y)+\\sqrt{t}\\right)\\preceq\\big(\\rho(x)\\rho(y)+t\\big){\\mathrm{e}}^{\\lambda_0\\frac{|x-y|^{2}}{t}}.\n\\end{align*}\nIn fact, using symmetry and the elementary inequality\n$$\n\\rho(x)\\leqslant \\rho(y)+|x-y|,\n$$\nwe have\n$$\n\\rho(x)^{2}+\\rho(y)^{2}\\preceq \\rho(x)\\rho(y)+|x-y|^{2}.\n$$\nThus, we can deduce that\n\\begin{align*}\n\\left(\\rho(x)+\\sqrt{t}\\right)\\left(\\rho(y)+\\sqrt{t}\\right)&\\preceq \\rho(x)\\rho(y)+t+\\rho(x)^{2}+\\rho(y)^{2}\\\\\n&\\preceq \\rho(x)\\rho(y)+t+|x-y|^{2}.\n\\end{align*}\nNote that for any  $\\lambda_0>0$, we have\n$$\n|x-y|^{2}\\preceq t\\cdot{\\mathrm{e}}^{\\lambda_0\\frac{|x-y|^{2}}{t}}.\n$$\nThe desired result follows immediately.\n\\end{proof}\n\nRecall the definition of $q_\\alpha(t,x,y)$ in (\\ref{q}). We give a better form of (\\ref{pd1ub})\nand \\eqref{pd1lb} as follows.\n\\begin{lemma}\nThere exist constants $\\lambda_1,\\lambda_2>0$, $C_1>1$ and $C_2<1$ such that\n\\begin{align}\np^{D}_{2}(t,x,y)&\\leqslant C_1q_2(t,x,y)\\xi^0_{\\lambda_1}(t,x-y), \\quad  (t, x, y)\\in (0, \\infty)\\times D\\times D,\\label{pd2ub}\\\\\np^{D}_{2}(t,x,y)&\\geqslant C_2q_2(t,x,y)\\xi^0_{\\lambda_2}(t,x-y), \\quad (t,x,y)\\in (0, 1]\\times D\\times D.\n\\label{pd2lb}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nThe lower bound \\eqref{pd2lb} is obvious, we only need to prove the upper bound \\eqref{pd2ub}. Combining (\\ref{pd1ub})\nand \\eqref{pd1lb} with (\\ref{pdd}), we have that for any $\\lambda_0>0$,\n\\begin{align*}\np^{D}_{2}(t,x,y)&\\preceq\n\\left(1\\wedge\\frac{\\rho(x)}{\\sqrt{t}}\\right)\\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^0_{\\lambda_0}(t,x-y).\n\\end{align*}\nThus, (\\ref{pd2ub}) is true when $|x-y|\\leqslant \\sqrt{t}$. On the other hand, notice that for $0<\\tilde{\\lambda}_0<\\lambda_0$ we have\n\\begin{align}\\label{new}\n\\frac{\\rho(x)}{\\sqrt{t}}{\\mathrm{e}}^{-\\lambda_0\\frac{|x-y|^{2}}{t}}=\\frac{\\rho(x)}{|x-y|}\\cdot\\frac{|x-y|}{\\sqrt{t}}{\\mathrm{e}}^{-\\lambda_0\\frac{|x-y|^{2}}{t}}\n\\preceq \\frac{\\rho(x)}{|x-y|}{\\mathrm{e}}^{-\\tilde{\\lambda}_0\\frac{|x-y|^{2}}{t}}.\n\\end{align}\nCombining \\eqref{new} with \\eqref{pd1ub} gives the desired result for $|x-y|>\\sqrt{t}$.\n\\end{proof}\n\n\n\nNow we prove the first and second order gradient estimates for\n$p^{D}_{2}(t,x,y)$.\n\n\\begin{lemma}\nLet $T>0$. There exist constants $C_T, \\lambda_3>0$ such that for $j=1,2$, \\\\\ni) for all $t\\in(0,T]$ and $x,y\\in D$,\n\\begin{align}\\label{p1}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|\\leqslant\nC_T\\hat q_2(t,y,x)\\xi^j_{\\lambda_3}(t,x-y);\n\\end{align}\nii) for all $t\\in(T,\\infty)$ and $x,y\\in D$,\n\\begin{align}\\label{p2}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|\\leqslant\\frac{C_T}{T^{j\/2}}\\hat q_2(t,y,x)\\xi^0_{\\lambda_3}(t,x-y),\n\\end{align}\nwhere $\\nabla^j_x$ denotes the $j$-order derivative with respect to the $x$ variable.\n\\end{lemma}\n\\begin{proof}\nFor (\\ref{p1}), we only need to show that there exist $\\lambda_3>0$ and $C_T>0$ such that for every $t\\in(0,T]$ and $x, y\\in D$,\n$$\n|\\nabla^j_xp^{D}_{2}(t,x,y)|\\leqslant C_T\\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^j_{\\lambda_3}(t,x-y).\n$$\nThen applying \\eqref{new}, we can get (\\ref{p1}).\nBy \\cite[VI.2, Theorem 2.1]{G-M}, we have that for every $t\\in(0,T]$ and $x, y\\in D$,\n$$\n|\\nabla^j_xp^{D}_{2}(t,x,y)|\\preceq \\xi^j_{\\lambda_3}(t,x-y).\n$$\nUsing the Chapman-Kolmogorov equation, we have\n\\begin{align*}\n|\\nabla^j_xp^{D}_{2}(t,x,y)|&\\leqslant \\int_D\\big|\\nabla^j_xp^{D}_{2}(t\/2,x,z)\\big|\\cdot p^{D}_{2}(t\/2,z,y){\\mathord{{\\rm d}}} z\\\\\n&\\preceq \\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\int_D\\xi^j_{\\lambda_3}(t\/2,x-z)\\xi^0_{\\lambda_1}(t\/2,z-y){\\mathord{{\\rm d}}} z\\\\\n&\\preceq \\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^j_{\\lambda_3}(t,x-y).\n\\end{align*}\nThus (\\ref{p1}) is valid. We now prove (\\ref{p2}). Similarly, it suffices to show that for every $t>T$ and $x, y\\in D$,\n$$\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|\\leqslant\n\\frac{C_T}{T^{j\/2}}\\left(1\\wedge\\frac{\\rho(y)}{\\sqrt{t}}\\right)\\xi^0_{\\lambda_2}(t,x-y).\n$$\nBy (\\ref{pd2ub}), (\\ref{p1}) and the Chapman-Kolmogorov equation, we have for $t>T$,\n\\begin{align}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|&\\leqslant\\int_{D}\\left|\\nabla^j_x p^{D}_{2}(T,x,z)\\right|\\cdot p^{D}_{2}(t-T,z,y){\\mathord{{\\rm d}}} z \\nonumber\\\\\n&\\leqslant C_T\\int_{{\\mathbb R}^{d}}\\xi^j_{\\lambda_2}(T,x-z)\\xi^0_{\\lambda_1}(t-T,z-y){\\mathord{{\\rm d}}} z\\nonumber\\\\\n&\\leqslant \\frac{C_T}{T^{j\/2}}\\xi^0_{\\lambda_2}(t,x-y).  \\label{gr0}\n\\end{align}\nFurthermore, for $t\\in(T,2T]$ the same argument yields that\n\\begin{align*}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|&\\leqslant\\int_{D}\\left|\\nabla^j_x p^{D}_{2}(t\/2,x,z)\\right|\\cdot p^{D}_{2}(t\/2,z,y){\\mathord{{\\rm d}}} z\\\\\n&\\leqslant C_T\\int_{{\\mathbb R}^d}\\xi^j_{\\lambda_2}(t\/2,x-z)\\frac{\\rho(y)}{\\sqrt{t\/2}}\\xi^0_{\\lambda_1}(t\/2,z-y){\\mathord{{\\rm d}}} z\\\\\n&\\leqslant \\frac{C_T}{T^{j\/2}}\\rho(y)\\xi^1_{\\lambda_2}(t,x-y)=\\frac{C_T}{T^{j\/2}}\\frac{\\rho(y)}{\\sqrt{t}}\\xi^0_{\\lambda_2}(t, x-y).\n\\end{align*}\nUsing (\\ref{gr0}) we get that for any $t\\in(2T,\\infty)$,\n\\begin{align*}\n\\left|\\nabla^j_x p^{D}_{2}(t,x,y)\\right|&\\leqslant\\int_{D}\\left|\\nabla^j_x p^{D}_{2}(t\/2,x,z)\\right|\\cdot p^{D}_{2}(t\/2,z,y){\\mathord{{\\rm d}}} z\\\\\n&\\leqslant \\frac{C_T}{T^{j\/2}}\\int_{{\\mathbb R}^d}\\xi^0_{\\lambda_2}(t\/2,x-z)\\frac{\\rho(y)}{\\sqrt{t\/2}}\\xi^0_{\\lambda_1}(t\/2,z-y){\\mathord{{\\rm d}}} z\\\\\n&\\leqslant \\frac{C_T}{T^{j\/2}}\\rho(y)\\xi^1_{\\lambda_2}(t,x-y)=\\frac{C_T}{T^{j\/2}}\\frac{\\rho(y)}{\\sqrt{t}}\\xi^0_{\\lambda_2}(t, x-y).\n\\end{align*}\nCombining the above computations, we get the\ndesired result.\n\\end{proof}\n\n\\begin{remark}\nIn fact, in the form of (\\ref{pd3}), our result means that for every $t\\in(0,T]$,\n$$\n|\\nabla^j_x p^D_{2}(t,x,y)|\\leqslant\nC_T\\frac{(|x-y|+\\sqrt{t})^{1-j}}{\\rho(x)\\wedge(|x-y|+\\sqrt{t})}q_2(t,x,y)\\xi^0_{\\lambda_3}(t,x-y).\n$$\nCompared with (\\ref{pd1ub}), (\\ref{pd1lb}) and (\\ref{pd3}),\nthe additional term $|x-y|$ in (\\ref{pd2ub})--(\\ref{pd2lb}) and (\\ref{p1})--(\\ref{p2}) is of critical importance\nin our derivation of the gradient estimates of $r^D(t,x,y)$ below.\n\\end{remark}\n\nRecall the definition of $q^D(t,x,y)$ in (\\ref{qq}). Now, we are ready to derive the following gradient estimates for the Dirichlet heat kernel\n$r^D(t,x,y)$.\n\n\\begin{lemma}\nLet $T>0$. There exists a constant $C_T>0$ such that for $j=1,2$,\nall $t\\in(0,T]$ and $x,y\\in D$,\n\\begin{align}\\label{grad}\n|\\nabla^j_x\nr^D(t,x,y)|\\leqslant C_T\\frac{(|x-y|+t^{1\/\\alpha})^{1-j}}{\\rho(x)\\wedge(|x-y|+t^{1\/\\alpha})}q^D(t,x,y).\n\\end{align}\nMoreover, for any $\\vartheta\\in(0,1)$ and $t\\in(0,T]$, $x,x',y\\in D$, we have\n\\begin{align}\n|\\nabla_x r^D(t,x,y)-\\nabla_x r^D(t,x',y)| \\leqslant C_T|x-x'|^\\vartheta \\hat q_{\\alpha}(t,y,\\widetilde x)\\varrho_{d+1+\\vartheta}^1(t,\\widetilde x-y), \\label{hold}\n\\end{align}\nwhere $\\widetilde x$ is the point among $x$ and $x'$ which is closer to $y$.\n\\end{lemma}\n\\begin{proof}\nWe claim that for $j=1,2$,\n\\begin{align}\n|\\nabla^j_x r^D(t,x,y)|\\preceq\n\\hat q_\\alpha(t,y,x)\\varrho^1_{d+j}(t,x-y).\\label{gr22}\n\\end{align}\nAs a consequence of this claim, we get\n\\begin{align*}\n|\\nabla^j_x r^D(t,x,y)|&\\preceq\\frac{1}{(|x-y|+t^{1\/\\alpha})^j\\hat q_\\alpha(t,x,y)} \\hat q_\\alpha(t,x,y)\\hat q_\\alpha(t,y,x)\\varrho_{d}^1(t,x-y)\\\\\n&\\asymp\\frac{(|x-y|+t^{1\/\\alpha})^{1-j}}{\\rho(x)\\wedge(|x-y|+t^{1\/\\alpha})}q^D(t,x,y).\n\\end{align*}\nNow we prove the claim (\\ref{gr22}).\nFrom \\cite[(4.1)]{Song}, we know that for all\n$\\xi\\in\\mathbb{R}^{d},$\n\\begin{align*}\n\\int_{0}^{\\infty}\\!s^{-d\/2}{\\mathrm{e}}^{-\\frac{|\\xi|^{2}}{s}}\\mu(t,s){\\mathord{{\\rm d}}}\ns\\asymp\\varrho^1_{d}(t,\\xi).\n\\end{align*}\nCombining this with (\\ref{rtxy}), (\\ref{p1}) and (\\ref{p2}), we can\nget\n\\begin{align}\n|\\nabla^j_x r^D(t,x,y)|&\\leqslant \\int_{0}^{1}\\left|\\nabla^j_x p^{D}_{2}(s,x,y)\\right|\\mu(t,s){\\mathord{{\\rm d}}} s+\\int_{1}^{\\infty}\\left|\\nabla^j_x p^{D}_{2}(s,x,y)\\right|\\mu(t,s){\\mathord{{\\rm d}}} s\\nonumber\\\\\n&\\preceq\\left(1\\wedge\\frac{\\rho(y)}{|x-y|}\\right)\\left[\\int_{0}^{\\infty}\\!\\xi^j_{\\lambda_3}(s,x-y)\\mu(t,s){\\mathord{{\\rm d}}} s\n+\\int_{0}^{\\infty}\\!\\xi^0_{\\lambda_3}(s,x-y)\\mu(t,s){\\mathord{{\\rm d}}} s\\right]\\nonumber\\\\\n&\\asymp\\left(1\\wedge\\frac{\\rho(y)}{|x-y|}\\right)\\Big[\\varrho^1_{d+j}(t,x-y)+\\varrho_{d}^1(t,x-y)\\Big]\\nonumber\\\\\n&\\preceq\\left(1\\wedge\\frac{\\rho(y)}{|x-y|}\\right)\\varrho^1_{d+j}(t,x-y),\n\\label{e1}\n\\end{align}\nwhere in the last inequality we have used the fact that $D$ is bounded and $t\\in(0,T]$. Thus, (\\ref{gr22}) is true when $|x-y|\\geqslant t^{1\/\\alpha}$. For the case that $|x-y|<t^{1\/\\alpha}$,\nwe may argue similarly to get that\n\\begin{align*}\n|\\nabla^j_x r^D(t,x,y)|\n&\\preceq \\rho(y)\\left[\\int_{0}^{\\infty}\\!\\xi^{j+1}_{\\lambda_3}(s,x-y)\\mu(t,s){\\mathord{{\\rm d}}} s\n+\\int_{0}^{\\infty}\\!\\xi^1_{\\lambda_3}(s,x-y)\\mu(t,s){\\mathord{{\\rm d}}} s\\right]\\\\\n&\\asymp\\rho(y)\\Big[\\varrho^1_{d+j+1}(t,x-y)+\\varrho_{d+1}^1(t,x-y)\\Big]\\\\\n&\\preceq\\frac{\\rho(y)}{t^{1\/\\alpha}}\\varrho^1_{d+j}(t,x-y).\n\\end{align*}\nThis together with estimate (\\ref{e1}) implies (\\ref{gr22}).\n\nFor (\\ref{hold}), without loss of generality, we may assume that $|x-y|\\leqslant |x'-y|$. Using (\\ref{gr22}) with $j=1$, we can get that when $|x-x'|\\geqslant (|x-y|+t^{1\/\\alpha})\/2$,\n\\begin{align*}\n{\\mathscr Q}&:=|\\nabla_x r^D(t,x,y)-\\nabla_x r^D(t,x',y)|\\\\\n&\\leqslant C_T|x-x'|^\\vartheta(|x-y|+t^{1\/\\alpha})^{-\\vartheta}\\Big(\\hat q_\\alpha(t,y,x)\\varrho^{1}_{d+1}(t,x-y)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\qquad\\quad+\\hat q_\\alpha(t,y,x')\\varrho^{1}_{d+1}(t,x'-y)\\Big)\\\\\n&\\leqslant C_T|x-x'|^\\vartheta\\hat q_\\alpha(t,y,x)\\varrho_{d+1+\\vartheta}^{1}(t,x-y).\n\\end{align*}\nWhen $|x-x'|<(|x-y|+t^{1\/\\alpha})\/2$, we have by the mean value theorem and (\\ref{gr22}) with $j=2$ that for some $\\varepsilon\\in[0,1]$,\n\\begin{align*}\n{\\mathscr Q}&\\leqslant C_T|x-x'|\\hat q_\\alpha\\big(t,y,x+\\varepsilon(x'-x)\\big)\\varrho^{1}_{d+2}\\big(t,x+\\varepsilon(x'-x)-y\\big)\\\\\n&\\leqslant C_T|x-x'|\\hat q_\\alpha(t,y,x)\\varrho_{d+2}^{1}(t, x-y)\\\\\n&\\leqslant C_T|x-x'|^\\vartheta\\hat q_\\alpha(t,y,x)\\varrho_{d+1+\\vartheta}^{1}(t,x-y).\n\\end{align*}\nThe proof is finished.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{main}}\n\n Let\n\\begin{align*}\n\\widehat{\\bf K}_{D}^{\\alpha-1}:=\\bigg\\{f\\in L^1_{loc}(D): \\lim_{t\\downarrow0}\\sup_{x\\in D}\\int_{D}&\\left(1\\wedge \\frac{\\rho(y)}{|x-y|}\\right)\\\\\n&\\times\\bigg(\\frac{1}{|x-y|^{d+1-\\alpha}}\\wedge\\frac{t^2}{|x-y|^{d+\\alpha+1}}\\bigg)|f(y)|{\\mathord{{\\rm d}}} y=0\\bigg\\}.\n\\end{align*}\nWe first give the following result about our Kato class.\n\n\\begin{lemma}\\label{include}\nWe have ${\\bf K}_D^{\\alpha-1}\\subset \\widehat{\\bf K}_D^{\\alpha-1}\\subset{\\mathbb K}_D^{0}$.\nMoreover, for any $\\gamma\\geqslant 0$, if $1<p,q\\leqslant \\infty$ satisfy\n\\begin{align}\n\\frac{d}{\\alpha p}+\\frac{1}{q}<1-\\frac{1+\\gamma}{\\alpha},    \\label{pq}\n\\end{align}\nthen $L^q({\\mathbb R};L^p(D))\\subseteq{\\mathbb K}_D^\\gamma$.\n\\end{lemma}\n\\begin{proof}\nIt follows from \\cite[Lemma 2.1]{C-S-X-X}, which follows from \\cite[Corollary 12]{Bo-Ja0},\nthat a real-valued function $f$ belongs to ${\\bf K}^{\\alpha-1}_D$ if and only if\n\\begin{equation*}\n\\lim_{t\\rightarrow0}\\sup_{x\\in D}\\int_{D}\\bigg(\\frac{1}{|x-y|^{d+1-\\alpha}}\\wedge\\frac{t^2}{|x-y|^{d+\\alpha+1}}\\bigg)|f(y)|{\\mathord{{\\rm d}}} y=0.\n\\end{equation*}\nThus the first inclusion is obvious.\nTo show that a real-valued time-independent function $f$ on $D$\nbelongs to ${\\mathbb K}_D^0$,\nit suffices to show that\n$$\n\\int_0^t\\varrho_{d+1}^1(s,x-y){\\mathord{{\\rm d}}} s\\preceq \\frac{1}{|x-y|^{d+1-\\alpha}}\\wedge\\frac{t^2}{|x-y|^{d+\\alpha+1}}.\n$$\nThis follows directly from \\cite[Lemma 2.3]{C-S-X-X} with $\\gamma=1$.\nThus the second inclusion is valid. Now we prove the third inclusion. By H\\\"older's inequality, we get\n$$\nK_f^\\gamma(\\delta)\\leqslant\\Bigg(\\int_{{\\mathbb R}}\\left(\\int_{D}|f(s,y)|^p\n{\\mathord{{\\rm d}}} y\\right)^{\\frac{q}{p}}{\\mathord{{\\rm d}}} s\\Bigg)^{\\frac{1}{q}}I_{\\alpha,\\gamma}(\\delta),\n$$\nwhere\n$$\nI_{\\alpha,\\gamma}(\\delta):=\\delta^{\\frac{\\gamma}{\\alpha}}\\Bigg(\\int^\\delta_0\\big[s^{-\\gamma\/\\alpha}+(\\delta-s)^{-\\gamma\/\\alpha}\\big]^{q^*}\\Bigg(\\!\\int_{{\\mathbb R}^d}\\frac{s^{p^*}}\n{\\big(|y|+s^{1\/\\alpha}\\big)^{(d+\\alpha+1)p^{*}}}{\\mathord{{\\rm d}}} y\\Bigg)^{\\frac{q^*}{p^*}}{\\mathord{{\\rm d}}} s\\Bigg)^{\\frac{1}{q^{*}}},\n$$\nwith $q^*:=\\frac{q}{q-1}$ and $p^*:=\\frac{p}{p-1}$. Noticing that\n\\begin{align*}\n\\int_{{\\mathbb R}^d}\\frac{s^{p^*}}{\\big(|y|+s^{1\/\\alpha}\\big)^{(d+\\alpha+1)p^{*}}}{\\mathord{{\\rm d}}} y\n&\\leqslant s^{p^*}\\left(\\int_{|y|\\leqslant s^{1\/\\alpha}}s^{-\\frac{(d+\\alpha+1)p^{*}}{\\alpha}}{\\mathord{{\\rm d}}} y\n+ \\int_{|y|> s^{1\/\\alpha}}\\frac{{\\mathord{{\\rm d}}} y}{|y|^{(d+\\alpha+1)p^{*}}}\\right)\\\\\n&\\preceq s^{\\frac{d-(d+1)p^{*}}{\\alpha}},\n\\end{align*}\nwe have\n$$\nI_{\\alpha,\\gamma}(\\delta)\n\\preceq\\delta^{\\frac{\\gamma}{\\alpha}}\\Bigg(\\int^\\delta_0\\big[s^{-\\gamma\/\\alpha}+(\\delta-s)^{-\\gamma\/\\alpha}\\big]^{q^*}s^{\\frac{dq^*}{\\alpha p^*}-\\frac{(d+1)q^*}{\\alpha}}{\\mathord{{\\rm d}}} s\\Bigg)^{\\frac{1}{q^{*}}}.\n$$\nThus $I_{\\alpha,\\gamma}(\\delta)$ converges to zero as $\\delta\\to 0$ provided that\n$$\n-\\frac{\\gamma q^*}{\\alpha}+\\frac{dq^*}{\\alpha p^*}-\\frac{d+1}{\\alpha}q^*+1>0\\Leftrightarrow (\\ref{pq}).\n$$\nThe desired result follows.\n\\end{proof}\n\n\nThe following lemma is related to the smallness of $b\\cdot\\nabla$ as a perturbation of $-(-\\Delta|_{D})^{\\alpha\/2}$, which plays an important role in proving our main result.\n\n\\begin{lemma}\\label{integral1}\nLet $\\delta>0$ and $b\\in {\\mathbb K}_{D}^{0}$. Then for all $0\\leqslant s<t\\leqslant s+\\delta$ and $x, y\\in\nD$, we have\n\\begin{align*}\n\\int_s^t\\!\\!\\!\\int_{D}r^D(r-s,x,z)|b(r,z)|\\cdot|\\nabla_z\nr^D(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r \\leqslant C(\\delta)r^D(t-s,x,y),\n\\end{align*}\nwhere $C(\\delta)$ is a positive constant with $C(\\delta)\\rightarrow0$ as $\\delta\\downarrow0$.\n\\end{lemma}\n\\begin{proof}\nIn this proof we always assume that $0\\leqslant s<t\\leqslant s+\\delta$ and $x, y\\in D$.\nFor brevity, we write\n$$\n{\\mathcal W}:=\\frac{r^D(r-s,x,z)|\\nabla_z r^D(t-r,z,y)|}{r^D(t-s,x,y)}.\n$$\nIt follows from (\\ref{grad}) that\n\\begin{align*}\n{\\mathcal W}&\\preceq \\frac{r^D(r-s,x,z)r^D(t-r,z,y)}{r^D(t-s,x,y)}\\cdot\\frac{1}{\\rho(z)\\wedge\\left(|z-y|+(t-r)^{1\/\\alpha}\\right)}\\\\\n&\\preceq \\frac{r^D(r-s,x,z)r^D(t-r,z,y)}{r^D(t-s,x,y)}\\left(\\frac{1}{\\rho(z)}+\\frac{1}{|z-y|+(t-r)^{1\/\\alpha}}\\right)=:{\\mathcal W}_1+{\\mathcal W}_2.\n\\end{align*}\nBy (\\ref{3r}), we have that\n\\begin{align}\\label{i}\n{\\mathcal W}_1&\\preceq \\big((r-s)\\wedge (t-r)\\big)\\Big([\\hat q_\\alpha(r-s,z,x)]^2\\varrho_{d}^0(r-s,x-z)\\nonumber\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad+[\\hat q_\\alpha(t-r,z,y)]^2\\varrho_{d}^0(t-r,z-y)\\Big)\\frac{1}{\\rho(z)}\\nonumber\\\\\n&\\preceq \\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)\n+\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y).\n\\end{align}\nAgain by (\\ref{3r}), we have\n\\begin{align*}\n{\\mathcal W}_2&\\preceq \\big((r-s)\\wedge (t-r)\\big)\\Big([\\hat q_\\alpha(r-s,z,x)]^2\\varrho_{d}^0(r-s,x-z)\\\\\n&\\qquad\\qquad\\qquad\\qquad+[\\hat q_\\alpha(t-r,z,y)]^2\\varrho_{d}^0(t-r,z-y)\\Big)\\frac{1}{|z-y|+(t-r)^{1\/\\alpha}}.\n\\end{align*}\nBy the same argument as in (\\ref{semi}), in the case $|x-z|+(r-s)^{1\/\\alpha}\\leqslant |z-y|+(t-r)^{1\/\\alpha}$, we have\n\\begin{align*}\n{\\mathcal W}_2&\\preceq [\\hat q_\\alpha(r-s,z,x)]^2\\varrho_{d}^1(r-s,x-z)\\frac{1}{|x-z|+(r-s)^{1\/\\alpha}}\\\\\n&\\preceq \\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z).\n\\end{align*}\nIn the case $|x-z|+(r-s)^{1\/\\alpha}>|z-y|+(t-r)^{1\/\\alpha}$, we have\n\\begin{align*}\n{\\mathcal W}_2&\\preceq  [\\hat q_\\alpha(t-r,z,y)]^2\\varrho_{d}^1(t-r,z-y)\\frac{1}{|z-y|+(t-r)^{1\/\\alpha}}\\\\\n&\\preceq \\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y).\n\\end{align*}\nHence,\n$$\n{\\mathcal W}_2\\preceq\n\\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)+\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y),\n$$\nwhich together with (\\ref{i}) yields that\n$$\n{\\mathcal W}\\preceq \\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)\n+\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y).\n$$\nConsequently, by the definition of Kato class\n${\\mathbb K}_{D}^{0},$ it holds that\n\\begin{align*}\n&\\quad\\int_s^t\\!\\!\\!\\int_{D}r^D(r-s,x,z)|b(r,z)|\\cdot|\\nabla_{z}\nr^D(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq\\int_s^t\\!\\!\\!\\int_{D}\\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)|b(r,z)|{\\mathord{{\\rm d}}}\nz{\\mathord{{\\rm d}}} r\\cdot r^D(t-s,x,y)\\\\\n&\\quad+\\int_s^t\\!\\!\\!\\int_{D}\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y)|b(r,z)|{\\mathord{{\\rm d}}}\nz{\\mathord{{\\rm d}}} r\\cdot r^D(t-s,x,y)\\\\\n&\\leqslant 2K_{b}^{0}(\\delta)r^D(t-s,x,y),\n\\end{align*}\nwhere $K_{b}^{0}(\\delta)$\nis defined in Definition \\ref{Definition}. The proof is thus finished.\n\\end{proof}\n\nTo derive the gradient estimate of the Dirichlet heat kernel, we shall also need the following\nresult.\n\n\\begin{lemma}\\label{integral2}\nLet $\\delta>0$ and $b\\in {\\mathbb K}_{D}^{0}$. Then for all $0\\leqslant s<t\\leqslant s+\\delta$ and $x, y\\in D$, we have\n\\begin{align*}\n&\\int_s^t\\!\\!\\!\\int_{D}|\\nabla_x r^D(r-s,x,z)||b(r,z)|\\cdot|\\nabla_zr^D(t-r,z,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant\n\\hat C(\\delta)\\frac{1}{\\rho(x)\\wedge(|x-y|+(t-s)^{1\/\\alpha})}r^D(t-s,x,y),\n\\end{align*}\nwhere $\\hat C(\\delta)$ is a positive constant with $\\hat C(\\delta)\\rightarrow0$ as $\\delta\\downarrow 0$.\n\\end{lemma}\n\\begin{proof}\nIn this proof we always assume that $0\\leqslant s<t\\leqslant s+\\delta$ and $x, y\\in D$.\nDefine\n\\begin{align*}\n{\\mathcal V}&:=\\frac{|\\nabla_{x}r^D(r-s,x,z)|\\cdot|\\nabla_{z}\nr^D(t-r,z,y)|}{\\hat q_\\alpha(t-s,y,x)\\varrho_{d+1}^1(t-s,x-y)}.\n\\end{align*}\nIt follows from (\\ref{gr22}) that\n$$\n{\\mathcal V}\\preceq {\\mathcal Q}\\cdot\\frac{\\varrho_{d+1}^1(r-s,x-z)\\varrho_{d+1}^1(t-r,z-y)}{\\varrho_{d+1}^1(t-s,x-y)},\n$$\nwhere\n\\begin{align*}\n{\\mathcal Q}&:=\\frac{\\hat q_\\alpha(r-s,z,x)\\hat q_\\alpha(t-r,y,z)}{\\hat q_\\alpha(t-s,y,x)}.\n\\end{align*}\nUsing \\eqref{new}, we get\n\\begin{align*}\n{\\mathcal Q}&\\asymp\\rho(z)\\cdot\\frac{\\rho(y)+|x-y|+(t-s)^{1\/\\alpha}}{(\\rho(z)+|x-z|+(r-s)^{1\/\\alpha})(\\rho(y)+|z-y|+(t-r)^{1\/\\alpha})}\\\\\n&\\preceq\\rho(z)\\cdot\\frac{\\rho(z)+|x-z|+(r-s)^{1\/\\alpha}+\\rho(z)+|z-y|+(t-r)^{1\/\\alpha}}{(\\rho(z)+|x-z|+(r-s)^{1\/\\alpha})(\\rho(z)+|z-y|+(t-r)^{1\/\\alpha})}\\\\\n&=\\frac{\\rho(z)}{\\rho(z)+|x-z|+(r-s)^{1\/\\alpha}}+\\frac{\\rho(z)}{\\rho(z)+|z-y|+(t-r)^{1\/\\alpha}}\\\\\n&\\asymp\n\\Big(\\hat q_{\\alpha}(r-s,z,x)+\\hat q_{\\alpha}(t-r,z,y)\\Big).\n\\end{align*}\nCombining this with (\\ref{333}), and by the same argument as in (\\ref{semi}), we further have that\n\\begin{align}\n{\\mathcal V}&\\preceq[(r-s)\\wedge (t-r)]\\Big(\\hat q_{\\alpha}(r-s,z,x)+\\hat q_{\\alpha}(t-r,z,y)\\Big)\\nonumber\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\times \\big(\\varrho^0_{d+1}(r-s,x-z)+\\varrho^0_{d+1}(t-r,z-y)\\big)\\nonumber\\\\\n&\\preceq \\hat q_\\alpha(r-s,z,x)\\varrho^1_{d+1}(r-s,x-z)\n+\\hat q_\\alpha(t-r,z,y)\\varrho^1_{d+1}(t-r,z-y).   \\label{3g}\n\\end{align}\nHence,\n\\begin{align*}\n&\\int_s^t\\!\\!\\!\\int_{D}|\\nabla_x r^D(r-s,x,z)||b(r,z)|\\cdot|\\nabla_zr^D(t-r,z,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq\nK_b^0(\\delta)\\hat q_\\alpha(t-s,y,x)\\varrho_{d+1}^1(t-s,x-y)\\\\\n&\\leqslant K_b^0(\\delta)\\frac{1}{\\rho(x)\\wedge(|x-y|+(t-s)^{1\/\\alpha})}r^D(t-s,x,y),\n\\end{align*}\nwhich yields the desired result. The proof is finished.\n\\end{proof}\n\nWe now proceed to solve the integral equation (\\ref{Duhamel}). For\nall $0\\leqslant s<t$ and $x,y\\in D$, set $r_0(s,x;t,y):=r^D(t-s,x,y)$,\nand define inductively that for $k\\geq1$,\n\\begin{align}\\label{induction}\nr_k(s,x;t,y):=\\int_s^t\\!\\!\\!\\int_Dr_{k-1}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr_{0}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n\\end{align}\nThe following result is an easy consequence of Lemma \\ref{integral1}\nand Lemma \\ref{integral2}.\n\n\\begin{lemma}\nLet $\\delta>0$ and $b\\in{\\mathbb K}^{0}_D$.\nThen there exits a constant $c_1>1$ such that for all $k\\geqslant 1$,\n$0\\leqslant s<t\\leqslant s+\\delta$ and $x,y\\in D$, we have\n\\begin{align}\\label{3.5}\n|r_k(s,x;t,y)|\\leqslant [c_1C(\\delta)]^{k}q^D(t-s,x,y)\n\\end{align}\nand\n\\begin{align}\\label{GK}\n|\\nabla_x r_k(s,x;t,y)|\\leqslant [c_1\\hat C(\\delta)]^{k}\\frac{1}{\\rho(x)\\wedge(|x-y|+(t-s)^{1\/\\alpha})}q^D(t-s,x,y),\n\\end{align}\nwhere $C(\\delta)$ and $\\hat C(\\delta)$ are the constants in Lemma\n\\ref{integral1} and Lemma \\ref{integral2}, respectively. Moreover,\nit holds that\n\\begin{align}\nr_k(s,x;t,y)=\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr_{k-1}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.    \\label{pkpk}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nWe first prove (\\ref{3.5}) by induction. By Lemma\n\\ref{integral1} and the definition of $q^D(t,x,y)$, we know that (\\ref{3.5}) holds for $k=1$. Now\nsuppose that it holds for  $k>1$. Then by definition and\nusing Lemmas \\ref{integral1} and \\ref{sharp}, we have\n\\begin{align*}\n|r_{k+1}(s,x;t,y)|\n&\\leqslant\\int_s^t\\!\\!\\!\\int_D|r_{k}(s,x;r,z)|\\cdot|b(r,z)|\\cdot|\\nabla_z\nr_{0}(r,z;t,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant\n[c_1C(\\delta)]^{k}\\int_s^t\\!\\!\\!\\int_Dr^{D}(s,x;r,z)|b(r,z)|\\cdot|\\nabla_z\nr_{0}(r,z;s,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant [c_1C(\\delta)]^{k+1}q^D(t-s,x,y).\n\\end{align*}\nFollowing the same argument with Lemma \\ref{integral1} replaced by\nLemma \\ref{integral2}, we can show that (\\ref{GK}) is true. We\nproceed to prove (\\ref{pkpk}). It is obvious that (\\ref{pkpk}) holds\nfor $k=1$. Suppose that it is true for $k>1$. Then, we have\nby (\\ref{induction}) and Fubini's theorem that\n\\begin{align*}\nr_{k+1}(s,x;t,y)&=\\int_s^t\\!\\!\\!\\int_Dr_{k}(s,x;r,z)b(r,z)\\cdot\\nabla_z r_{0}(r,z;s,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&=\\int_s^t\\!\\!\\!\\int_D\\int_s^r\\!\\!\\!\\int_Dr_{0}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'} r_{k-1}(r',z';r,z){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\times b(r,z)\\cdot\\nabla_z r_{0}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&=\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r',z')b(r',z')\\cdot\\!\\int_{r'}^{t}\\!\\!\\int_D\\nabla_{z'} r_{k-1}(r',z';r,z)\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\times b(r,z)\\cdot\\nabla_z r_{0}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r{\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'\\\\\n&=\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'}r_{k}(r',z';t,y){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'.\n\\end{align*}\nThe proof is complete.\n\\end{proof}\n\nNow, we are ready to give:\n\n\\begin{proof}[Proof of Theorem \\ref{main}]\nLet $r_k$ be defined by \\eqref{induction}. For $\\delta>0$, define\n${\\mathbb D}_\\delta:=\\{(s,x;t,y): x,y\\in D\\,\\, \\text{and}\\,\\,0\\leqslant s<t\\leqslant s+\\delta\\}$.\nIt follows from Lemma \\ref{integral1} that there exists a\n$\\delta_0\\in (0, 1]$ such that for all $0\\leqslant s<t\\leqslant s+\\delta_0$, we have\n$c_1C(\\delta_0)<1\/4$,\nwhere $c_1$ and $C(\\delta_0)$ are the constants from Lemma \\ref{induction}. Hence,\n\\begin{align}\n\\sum_{k=0}^{\\infty}|r_{k}(s,x;t,y)|\\leqslant\n\\frac43 q^{D}(t-s,x,y)\\,\\,\\,\\text{on}\\,\\,\\,{\\mathbb D}_{\\delta_0},   \\label{es}\n\\end{align}\nwhich means that the series $\\sum_{k=0}^{\\infty}r_{k}(s,x;t,y)$\nconverges on ${\\mathbb D}_{\\delta_0}$. Define\n$r^{D,b}(s,x;t,y):=\\sum_{k=0}^{\\infty}r_{k}(s,x;t,y)$ on ${\\mathbb D}_{\\delta_0}$.\nBy \\eqref{induction}, we have\n\\begin{equation*}\n\\sum_{k=0}^{n+1}r_k(s,x;t,y)=r_0(s,x;t,y)+\\int_s^t\\!\\!\\!\\int_D\\sum_{k=0}^nr_k(s,x;r,z)b(r,z)\\cdot\\nabla_z r_0(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n\\end{equation*}\nLetting $n\\rightarrow\\infty$ on both sides, we get (\\ref{Duhamel}).\n\n\\vspace{2mm} \\noindent(i) The upper bound on ${\\mathbb D}_{\\delta_0}$ follows by (\\ref{es}). As\nfor the lower bound on ${\\mathbb D}_{\\delta_0}$, we have\n\\begin{align*}\nr^{D,b}(s,x;t,y)\\geqslant r^D(t-s,x,y)-\\sum_{k=1}^\\infty |r_k(s,x;t,y)|\\geqslant\n\\frac{2}{3}r^D(t-s,x,y).\n\\end{align*}\nThus, (\\ref{estimate}) is valid on ${\\mathbb D}_{\\delta_0}$.\n\nNow let $\\tilde{r}^{D,b}(s,x;t,y)$ be another solution to\nthe integral equation (\\ref{Duhamel}) satisfying (\\ref{estimate}) on ${\\mathbb D}_{\\delta_0}$.\nWe claim that for every $k\\in \\mathbb{N}$, there exists a constant $C_0$ such that\non ${\\mathbb D}_{\\delta_0}$,\n\\begin{align}\\label{unique}\n|r^{D,b}(s,x;t,y)-\\tilde{r}^{D,b}(s,x;t,y)|\\leqslant C_0[c_1C(\\delta_0)]^{k}q^{D}(t-s,x,y).\n\\end{align}\nIndeed, for $k=1$, using (\\ref{Duhamel}), (\\ref{estimate}) and Lemma\n\\ref{integral1} we have\n\\begin{align*}\n&\\quad|r^{D,b}(s,x;t,y)-\\tilde{r}^{D,b}(s,x;t,y)|\\\\\n&\\leqslant\\int_{s}^{t}\\!\\!\\!\\int_{D}\\big(|r^{D,b}(s,x;r,z)|+|\\tilde{r}^{D,b}(s,x;r,z)|\\big)\\cdot|b(r,z)|\\cdot|\\nabla_z\nr^{D}(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant C_0\\int_{s}^{t}\\!\\!\\!\\int_{D}r^{D}(r-s,x,z)\\cdot|b(r,z)|\\cdot|\\nabla_z\nr^{D}(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\leqslant C_0c_1C(\\delta_0)q^{D}(t-s,x,y).\n\\end{align*}\nSuppose that \\eqref{unique} holds for some $k\\in \\mathbb{N}$. By\n(\\ref{Duhamel}), Lemma \\ref{integral1} and the induction hypothesis,\nwe have\n\\begin{align*}\n&\\quad|r^{D,b}(s,x;t,y)-\\tilde{r}^{D,b}(s,x;t,y)|\\\\\n&\\leqslant\\int_{s}^{t}\\!\\!\\!\\int_{D}|r^{D,b}(s,x;r,z)-\\tilde{r}^{D,b}(s,x;r,z)|\\cdot|b(r,z)|\\cdot|\\nabla_z\nr^D(r,z;t,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant\nC_0[c_1C(\\delta_0)]^{k}\\int_{s}^{t}\\!\\!\\!\\int_{D}r^D(r-s,x,z)\\cdot|b(r,z)|\\cdot|\\nabla_z\nr^D(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\leqslant C_0[c_1C(\\delta_0)]^{k+1}q^D(t-s,x,y).\n\\end{align*}\nSince $c_1C(\\delta_0)<1$, letting $k\\rightarrow\\infty$ we obtain the\nuniqueness.\n\n\\vspace{2mm} \\noindent(ii)\nBy choosing $\\delta_0$ smaller if necessary, we can assume that $c_1\\hat C(\\delta_0)<1$\nfor $0\\leqslant s<t\\leqslant s+\\delta_0$,\nwhere $c_1$ and $\\widehat{C}(\\delta_0)$ are the constants from Lemma \\ref{induction}.\nIt then follows from \\eqref{GK} that on ${\\mathbb D}_{\\delta_0}$,\n\\begin{align*}\n\\left|\\sum_{k=0}^{\\infty}\\nabla_x\nr_{k}(s,x;t,y)\\right|\\preceq\\frac{1}{\\rho(x)\\wedge(|x-y|+(t-s)^{1\/\\alpha})}q^D(t-s,x,y),\n\\end{align*}\nwhich means that (\\ref{vi}) is true. Moreover, by (\\ref{pkpk}) and\nFubini's theorem, we have\n\\begin{align*}\n&\\quad r^{D,b}(s,x;t,y)=\\sum_{k=0}^{\\infty}r_{k}(s,x;t,y)\\\\\n&=r^D(s,x;t,y)+\\sum_{k=0}^{\\infty}\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr_{k}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&=r^D(s,x;t,y)+\\int_s^t\\!\\!\\!\\int_Dr_{0}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr^{D,b}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n\\end{align*}\nThis yields (\\ref{duhamel}).\n\n\\vspace{2mm} \\noindent(iii) By Fubini's theorem, we have\n$$\n\\int_Dr^{D,b}(s,x;r,z)r^{D,b}(r,z;t,y){\\mathord{{\\rm d}}}\nz=\\sum_{n=0}^{\\infty}\\sum_{m=0}^n\\int_Dr_m(s,x;r,z)r_{n-m}(r,z;t,y){\\mathord{{\\rm d}}}\nz.\n$$\nThus, for proving (\\ref{eq21}), it suffices to show that for each\n$n\\in{\\mathbb N}_0$,\n\\begin{align}\n\\sum_{m=0}^n\\int_Dr_{m}(s,x;r,z)r_{n-m}(r,z;t,y){\\mathord{{\\rm d}}} z=r_n(s,x;t,y).\n\\label{ck}\n\\end{align}\nIt is clear that the above equality holds for $n=0$. Suppose now\nthat it holds for some $n\\in{\\mathbb N}$. Write\n$$\n\\sum_{m=0}^{n+1}\\int_Dr_{m}(s,x;r,z)r_{n+1-m}(r,z;t,y){\\mathord{{\\rm d}}}\nz={\\mathcal J}_1+{\\mathcal J}_2,\n$$\nwhere\n$$\n{\\mathcal J}_1:=\\int_Dr_{n+1}(s,x;r,z)p_{0}(r,z;t,y){\\mathord{{\\rm d}}} z\n$$\nand\n$$\n{\\mathcal J}_2:=\\sum_{m=0}^{n}\\int_Dr_{m}(s,x;r,z)p_{n+1-m}(r,z;t,y){\\mathord{{\\rm d}}} z.\n$$\nBy (\\ref{induction}) and Fubini's theorem, we have\n\\begin{align*}\n{\\mathcal J}_{1}&=\\int_{D}\\left(\\int_{s}^{r}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\n\\nabla_{z'} r_{0}(r',z';r,z){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'\\right)r_{0}(r,z;t,y){\\mathord{{\\rm d}}} z\\\\\n&=\\int_{s}^{r}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\\left(\\int_{D}\\nabla_{z'}\nr_{0}(r',z';r,z)r_{0}(r,z;t,y){\\mathord{{\\rm d}}} z\\right){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'\\\\\n&=\\int_{s}^{r}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'}\nr_{0}(r',z';t,y){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'.\n\\end{align*}\nSimilarly, by (\\ref{induction}) and the induction hypothesis, we\nhave\n\\begin{align*}\n{\\mathcal J}_{2}=\\int_{r}^{t}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'}\nr_{0}(r',z';t,y){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'.\n\\end{align*}\nHence,\n\\begin{align*}\n{\\mathcal J}_{1}+{\\mathcal J}_{2}=\\int_{s}^{t}\\!\\!\\!\\int_{D}r_{n}(s,x;r',z')b(r',z')\\cdot\\nabla_{z'}\nr_{0}(r',z';t,y){\\mathord{{\\rm d}}} z'{\\mathord{{\\rm d}}} r'=r_{n+1}(s,x;t,y),\n\\end{align*}\nwhich gives (\\ref{ck}).\n\n\\vspace{2mm}\nNow, we can extend  $r^{D,b}(s,x;t,y)$ from ${\\mathbb D}_{\\delta_0}$ to\n$\\{(s,x;t,y): x,y\\in D\\,\\, \\text{and}\\,\\,0\\leqslant s<t<\\infty\\}$.\nThen it is routine to\nextend the above assertions (i)-(iii) on ${\\mathbb D}_{\\delta_0}$ to\n$D_{\\delta}$ for any$\\delta>0$.\n\n\\vspace{2mm} \\noindent(iv) Let $R_{s,t}f(x):=\\int_Dr^D(t-s,x,y)f(y){\\mathord{{\\rm d}}} y$.\nBy (\\ref{Duhamel}), we have for any $f\\in C_c^2(D)$,\n\\begin{align}\\label{RT5}\nR_{s,t}^{D,b}f(x)=R_{s,t}f(x)+\\int^t_s R_{s,r}^{D,b}(b\\cdot\\nabla R_{r,t}f)(x){\\mathord{{\\rm d}}} r.\n\\end{align}\nIt then follows that\n\\begin{align}\nR_{s,t}^{D,b}f(x)-f(x)&=R_{s,t}f(x)-f(x)+\\int^t_s R_{s,r}^{D,b}(b\\cdot\\nabla R_{r,t}f)(x){\\mathord{{\\rm d}}} r\\nonumber\\\\\n&=\\int_s^t R_{s,r}(-(-\\Delta|_D)^{\\alpha\/2})f(x){\\mathord{{\\rm d}}} r+\\int^t_s R_{s,r}^{D,b}(b\\cdot\\nabla R_{r,t}f)(x){\\mathord{{\\rm d}}} r,\\label{RT4}\n\\end{align}\nand, by \\eqref{RT5} and Fubini's theorem,\n\\begin{align*}\n&\\quad\\int_{s}^{t}R^{D,b}_{s,r}(-(-\\Delta|_D)^{\\alpha\/2})f(x){\\mathord{{\\rm d}}}\ns-\\int_s^t R_{s,r}(-(-\\Delta|_D)^{\\alpha\/2})f(x){\\mathord{{\\rm d}}} r\\\\\n&=\\int_{s}^{t}\\!\\!\\int_{s}^{r}R^{D,b}_{s,u}\\big(b\\cdot\\nabla R_{u,r}(-(-\\Delta|_D)^{\\alpha\/2})f\\big)(x){\\mathord{{\\rm d}}} u{\\mathord{{\\rm d}}} r\\\\\n&=\\int_{s}^{t}R^{D,b}_{s,u}b\\cdot\\nabla\\left(\\int_{u}^{t}R_{u,r}(-(-\\Delta|_D)^{\\alpha\/2})f(x){\\mathord{{\\rm d}}} r\\right){\\mathord{{\\rm d}}} u\\\\\n&=\\int_{s}^{t}R^{D,b}_{s,u}b\\cdot\\nabla\\Big(R_{u,t}f(x)-f(x)\\Big){\\mathord{{\\rm d}}} u.\n\\end{align*}\nCombining this with \\eqref{RT4}, we obtain\n\\begin{align*}\nR_{s,t}^{D,b}f(x)-f(x)=\\int_s^tR_{s,r}^{D,b}{\\mathscr L}^{D,b}f(x){\\mathord{{\\rm d}}}\nr,\n\\end{align*}\nwhich gives (\\ref{eq23}).\n\n\\vspace{2mm} \\noindent(v) Since $r^D(t,x,y)$ is the transition density\nof the process $Y^D$, so we have for any uniformly continuous function\n$f(x)$ with compact supports,\n\\begin{align*}\n\\lim_{t\\downarrow s}\\|R_{s,t}f-f\\|_\\infty=0.\n\\end{align*}\nMeanwhile, by (\\ref{estimate}) and Lemma \\ref{integral1} we have\n\\begin{align*}\n&\\quad\\left|\\!\\int_{D}\\!\\!\\!\\left(\\int_{s}^{t}\\!\\!\\!\\int_{D}r^{D,b}_{\\alpha}(s,x;r,z)b(r,z)\\cdot\\nabla_z\nr^D(t-r,z,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\!\\right)\\!f(y){\\mathord{{\\rm d}}} y\\right|\\\\\n&\\preceq\n\\|f\\|_{\\infty}\\int_{D}\\!\\!\\!\\left(\\int_{s}^{t}\\!\\!\\!\\int_{D}r^D(r-s,x,z)|b(r,z)|\\cdot|\\nabla_z\nr^D(t-r,z,y)|{\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\!\\right){\\mathord{{\\rm d}}} y\\\\\n&\\leqslant C(\\delta)\\|f\\|_{\\infty}\\int_{D}r^D(t-s,x,y){\\mathord{{\\rm d}}} y\\leqslant\nC(\\delta)\\|f\\|_{\\infty},\n\\end{align*}\nwhich yields (\\ref{con}) by (\\ref{Duhamel}).\n\n\\vspace{2mm} \\noindent(vi) Set\n$$\n\\Phi(s,x;t,y):=\\int_s^t\\!\\!\\!\\int_D r^{D}(r-s,x,z)b(r,z)\\cdot\\nabla_z r^{D,b}(r,z;t,y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r.\n$$\nIf we further assume that for $\\gamma\\in(0,\\alpha-1)$, $b\\in{\\mathbb K}_D^{\\gamma}$, then using (\\ref{hold}) we have for any $x,x',y\\in D$,\n\\begin{align*}\n&|\\Phi(s,x;t,y)-\\Phi(s,x';t,y)|\\preceq |x-x'|^{\\gamma}\\int_s^t\\!\\!\\!\\int_D\\hat q_{\\alpha}(r-s,z,\\tilde x)\\varrho_{d+1+\\gamma}^1(r-s,\\tilde x-z)\\\\\n&\\qquad\\qquad\\times|b(r,z)|\\hat q_{\\alpha}(t-r,y,z)\\varrho_{d+1}^1(t-r,z-y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq |x-x'|^{\\gamma}\\int_s^t\\!\\!\\!\\int_D(r-s)^{-\\gamma\/\\alpha}\\hat q_{\\alpha}(r-s,z,\\tilde x)\\varrho_{d+1}^1(r-s,\\tilde x-z)\\\\\n&\\qquad\\qquad\\times|b(r,z)|\\hat q_{\\alpha}(t-r,y,z)\\varrho_{d+1}^1(t-r,z-y){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq |x-x'|^{\\gamma}\\hat q_{\\alpha}(t-s,y,\\tilde x)\\varrho_{d+1}^1(t-s,\\tilde x-y)\\int_s^t\\!\\!\\!\\int_D(r-s)^{-\\gamma\/\\alpha}|b(r,z)|\\\\\n&\\qquad\\qquad\\times\\Big(\\hat q_{\\alpha}(r-s,z,\\tilde x)\\varrho_{d+1}^1(r-s,\\tilde x-z)+\\hat q_{\\alpha}(t-r,z,y)\\varrho_{d+1}^1(t-r,z-y)\\Big){\\mathord{{\\rm d}}} z{\\mathord{{\\rm d}}} r\\\\\n&\\preceq |x-x'|^{\\gamma}(t-s)^{-\\gamma\/\\alpha}\\hat q_{\\alpha}(t-s,y,\\tilde x)\\varrho_{d+1}^1(t-s,\\tilde x-y),\n\\end{align*}\nwhere the third inequality is due to (\\ref{3g}), and the last inequality follows from the definition of ${\\mathbb K}_D^\\gamma$. Combining this with (\\ref{duhamel}) and (\\ref{hold}), we get the desired result. The proof is complete.\n\\end{proof}\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:introduction}\nAs a result of hierarchically-structured, accreting molecular clouds, stars preferentially form in clustered environments. More than 70 \\% of massive stars are located in clusters and OB associations \\citep{ladalada03}, making young star clusters (YSCs) preferential sites from which radiative and mechanical feedback originates. Typically, only a minority of these stellar aggregates are gravitationally bound and will, therefore, survive as star clusters \\citep{kruijssen12a, grudic20}. These are the clusters that sit in the densest regions of the hierarchy and are therefore able to survive the removal of gas long enough to reach a high star formation efficiency and become dynamically relaxed and well-mixed \\citep{kruijssen19}. The remaining loosely-bound clusters will disperse in short timescales and blend into the field stellar population.\n\n\\par A detailed understanding of the process of stellar feedback is important as this mechanism leads to the suppression of global star formation in galaxies and to a multiphase and metal-enriched interstellar medium (ISM).\nIn particular, it will shed light on the origin of the diffuse ionised gas (DIG) that has been observed to constitute up to 50\\% of the total H$\\alpha$ luminosity in local spiral galaxies \\citep{ferguson96, hoopes96, zurita00, thilker02, oey07}. The origin of this gas is unclear \\citep[see e.g.][for a review]{mathis00, haffner09}, and it has been linked to various processes including leaking \\ion{H}{ii} regions~\\citep{zurita02, weilbacher18}, evolved field stars \\citep{hoopes00,zhang17}, shocks~\\citep{collins01}, and cosmic rays~\\citep{vandenbroucke18}.\nBoth theory and observation have shown that stellar feedback can facilitate the escape of ionising radiation from the star forming regions \\citep[see e.g.][]{bik15,menacho19} if the youngest star clusters have had time\nto clear out channels through the dense leftover gas from the parent giant molecular clouds (GMCs) from which they were born~\\citep{dale15,howard18}.\nIn order to study how the radiation from massive stars affects the ISM, it is therefore necessary to investigate the timescale for photons to find a channel through which they can leave the  region.\nVarious studies have found that small scale physics plays an important role in the escape of ionising radiation. In particular, the effect of temporal and spatial clustering of supernovae (SN) can have important consequences \\citep{gentry17,kim17,fielding18}, resulting in a transferred momentum up to an order of magnitude larger than for isolated SN.\nSimulations of individual galaxies do indeed indicate that the escape of ionising radiation is largely driven by the distribution of dense gas \\citep[see e.g.][]{paardekooper11}, and high-resolution GMC simulations have proven that photons are more likely to escape from less massive GMCs \\citep{howard18}. Consequently, late-emitted ionising photons are more likely to escape the star forming region as the gas surrounding the region is dispersed by ionising radiation and supernova feedback~\\citep{kim13a}. \nThe same trend has been recently observed in high-resolution cosmological simulations, showing that photons are preferentially leaking from star forming regions containing feedback-driven superbubbles~\\citep{ma20}.\nFinally, stars are also responsible of yielding metals that will enrich the ISM, and a detailed understanding of abundance variations in star forming regions is needed in order to constrain chemical evolution models. However, until recently, it has been unclear whether star forming galaxies feature, in general, a homogeneous metal enrichment or whether the enrichment varies in different star forming regions and within the regions themselves.\nRecent high-resolution studies \\citep[e.g.][]{james16, mcleod19} seem to indicate the latter, implying that feedback mechanisms might play an important role in local metallicity variations.\n\n\\par High resolution studies of star forming regions are therefore an essential piece of information in order to bridge the spatial scales that critically connect the sources of feedback (stars and clusters, i.e. a few parsec scales) and their immediate surroundings (10s parsec scale) to galactic scale dynamics (kpc scale).\nSuch studies, initially limited to narrow-band photometry \\citep[e.g.][]{pellegrini12}, were revolutionised with the advent of instruments with high spatial resolution integral field spectroscopy capabilities, such as the Multi Unit Spectroscopic Explorer~\\citep[MUSE,][]{bacon10} instrument at ESO Very Large Telescope (VLT). Recently, \\citet{mcleod19,mcleod20} exploited the MUSE capability for a detailed study of \\ion{H}{ii} regions in the Large Magellanic Cloud and in the dwarf spiral NGC 300, mapping the ionisation structure of the regions and constraining their escape of ionising radiation.\nIn this work, we perform a similar study in the local spiral galaxy NGC 7793 at a spatial resolution of $\\sim$ 10 pc with MUSE. In \\citet[][henceforth Paper~\\textsc{i}]{paperI} we presented the data, constructed a  sample of \\ion{H}{ii} regions, determined the fraction of DIG in our field of view (FoV) and studied the properties of the ionised gas in emission line diagrams.\nHere, we combine the MUSE results on the ionised gas with the stellar and cluster population studied with the Hubble Space Telescope (HST) and the molecular gas mapped with the Atacama Large Millimeter\/submillimeter Array (ALMA),\nconstruct a photoionisation budget for each \\ion{H}{ii} region and study how it relates to its stellar and GMC content and to its ionisation structure.\n\n\\par This work is structured as follows.\nIn Sect.~\\ref{section:data} we give a brief overview of the data.\nIn Sect.~\\ref{section:stellar_pop} we present the stellar census in our FoV and analyse the distribution of YSC properties.\nIn Sect.~\\ref{section:extinction} we compare extinction maps derived from individual stellar extinction and from the ionised gas and in Sect.~\\ref{section:metallicity} we derive the oxygen abundance of the \\ion{H}{ii} regions.\nIn Sect.~\\ref{section:optical_depth} we analyse the ionisation structure of the gas, and in Sect.~\\ref{section:budget} we compile a photoionisation budget for the entire FoV and in more detail for a subset of the \\ion{H}{ii} regions.\nFinally, in Sect.~\\ref{section:discussion} we discuss our findings and in Sect.~\\ref{section:conclusions} we give a brief summary and present our conclusions.\n\n\\begin{figure*}\n\\center\n\\includegraphics[width=16cm]{stellar_census.pdf}\n\\caption{Stellar census in the MUSE FoV. The location of candidate O stars (white circles), YSCs (green triangles) and WR stars (blue crosses) is indicated, overlaid on a map of the H$\\alpha$ emission. Black contours indicate the \\ion{H}{ii} regions identified in Paper~\\textsc{i}, light blue filled contours denote the position of GMCs.}\n\\label{fig:stellar_census}\n\\end{figure*}\n\n\n\\section{Data description}\n\\label{section:data}\nThe data and data reduction steps are described in detail in Paper~\\textsc{i}: we give here a brief summary. We observed the nearby flocculent spiral galaxy NGC 7793 with MUSE\\footnote{ESO programme 60.A-9188(A), PI Adamo.}.\nThe dataset consists of two pointings in the Wide Field Mode (WFM) adaptive optics assisted (AO) configuration, with the extended wavelength setting. On the final datacube, we measure a PSF of $0.62 \\arcsec$ FWHM (at 7000~\\AA{}), corresponding to a spatial resolution of $\\sim$ 10 pc at the distance of NGC 7793.\n\\par NGC 7793 has also been observed in the framework of the HST Treasury programme LEGUS\\footnote{HST GO\u201313364.}~\\citep{calzetti15} and in ALMA CO(J = $2-1$).\nThe catalogues of YSCs derived from the LEGUS data are publicly available\\footnote{\\url{https:\/\/archive.stsci.edu\/prepds\/legus\/dataproducts-public.html}}: in this work, we make use of the catalogue generated with the Geneva stellar evolution models and Milky Way extinction. We are moreover selecting only clusters labelled as `Class 1' (compact and symmetric), `Class 2' (concentrated but with some degree of asymmetry), and `Class 3' (multiple peak systems). For more details on the cluster catalogues (e.g. cluster identification, classification and photometry) see~\\citet{adamo17}. \nLee et al. (in preparation) have identified O stars based on HST photometry. The catalogue is described in \\citet{wofford20}, and consists of\nmain sequence O stars of $M > 20~M_\\odot$, selected in a colour-magnitude and colour-$Q$ diagram\\footnote{Here, $Q$ is the reddening-free parameter introduced by \\citet{johnson53}.}.\nThe resulting completeness of this stellar catalogue is estimated to be about 75\\% at the lower mass limit of $20~M_\\odot$ and rapidly growing to 90\\% at stellar masses above $25~M_\\odot$\\footnote{We note that the mass of the candidates is not constrained by the selection method. However, mass estimates have been obtained from completeness simulations, in which the candidates are compared to synthetic stars.}. The authors report also an average contamination rate $\\sim$ 30\\%. We use the latter rate to account for uncertainties in the contribution of O stars to the ionisation budget.\n\nThe catalogue of GMCs built on the ALMA dataset is presented in \\citet{grasha18}. The ALMA data have an angular resolution of $0.85''$ and a velocity resolution of 1.2 km s$^{-1}$. The data are sensitive to emission up to $11''$ and are mapping GMC down to masses $\\sim 10^4 M_{\\sun}$.\n\n\\section{Properties of the stellar population}\n\\label{section:stellar_pop}\n\n\\begin{figure*\n\\center\n\\includegraphics[width=18cm]{ysc_age_mass_FOV.pdf}\n\\caption{Spatial distribution of YSC ages (\\textit{left panel}) and masses (\\textit{right panel}) in the MUSE FoV, overlaid on the H$\\alpha$ map shown in Fig.~\\ref{fig:stellar_census}. Circles and diamonds indicate, respectively, clusters younger and older than 10 Myr.\nAges and masses are best estimates computed from the median of the corresponding PDF (see Sect.~\\ref{section:ysc_properties}).\nBlack contours indicate the \\ion{H}{ii} regions identified in Paper~\\textsc{i}.}\n\\label{fig:ysc_distr_FOV}\n\\end{figure*}\n\n\\begin{figure*\n\\center\n\\includegraphics[width=17cm]{ysc_agemassq0_hist.png}\n\\caption{Posterior probability distributions of YSC age (\\textit{left}) mass (\\textit{centre}) and ionising flux (\\textit{right}) for clusters located in \\ion{H}{ii} regions (in blue) and for field clusters (in orange). Thick lines indicate the combined PDFs, thin lines correspond to the PDFs of single clusters. The grey dashed lines indicate the location of the main modes of the combined distributions.}\n\\label{fig:ysc_agemass_hist}\n\\end{figure*}\n\n\\begin{figure\n\\center\n\\includegraphics[width=7cm]{mass_distr_gmc_ysc.pdf}\n\\caption{Distribution of YSC mass (clusters with $t < 10$ Myr, in blue), GMC mass (in green) and \\ion{H}{ii} regions luminosity (in black, reddening corrected) in our FoV. YSC masses are best estimates computed from the median of the corresponding PDF (see Sect.~\\ref{section:ysc_properties}). The histograms are normalised to a peak value of 1.}\n\\label{fig:mass_distr_gmc_ysc}\n\\end{figure}\n\n\\subsection{Stellar census}\nWe present here the full census of the stellar population in our FoV. Figure~\\ref{fig:stellar_census} shows a map of the H$\\alpha$ emission line obtained from the integration of the MUSE stellar continuum subtracted datacube in the rest-frame wavelength range $6559 - 6568$~\\AA{}. Overlaid on the H$\\alpha$ emission, we show the position of candidate O stars from Lee et al. (in prep., white circles), YSCs from LEGUS (green triangles) and of Wolf Rayet (WR) stars that we spectroscopically identified in Paper~\\textsc{i} (listed in Table~\\ref{table:wr_stars}; blue crosses).\nWe also show the contours of \\ion{H}{ii} regions identified in Paper~\\textsc{i} (in black, corresponding to a flux brightness cut in H$\\alpha$ = $\\SI{6.7e-18}{\\erg \\per \\second \\per \\centi \\metre \\squared}$spaxel$^{-1}$) and of GMCs (light blue filled contours). \nThe total stellar census in our FoV consists of 651 O stars, 93 YSC, and 9 WR stars.\n\nFrom Fig.~\\ref{fig:stellar_census}, we observe that both YSCs and candidate O stars are abundant over the entire FoV. However, we note that several of the smaller and medium-bright \\ion{H}{ii} regions do not seem to host any cluster. Overall, we find\n$\\sim$ 49\\% field\\footnote{Throughout the paper, we refer to stars and clusters located outside \\ion{H}{ii} regions as `field' objects.} clusters (35\\% if considering only clusters with median age $<$ 10 Myr, as motivated in Sect.~\\ref{section:fesc_individual}) and\n$\\sim$ 38\\% field candidate O stars. \nWe note that the numbers of O-stars quoted here are not corrected for completeness, so that the latter fraction could in part be due to the high contamination rate of catalogue (see Sect.~\\ref{section:data}).\nWe observe that WR stars are populating the \\ion{H}{ii} regions, with the exception of one object (located in the upper left corner and catalogued as WR 3 in Table~\\ref{table:wr_stars}). \\citet{rate20} recently found that isolated WR stars are not an exception, and amount to at least 64\\% in the Milky Way \\citep[as observed in][]{gaia_DR2}. The authors explore the possible origin of field WR stars in simulations, finding that the most frequent mechanisms are the formation in a low density association that then expands during the star lifetime and the ejection from a cluster through dynamical interaction or binary disruption. This has been observed in galactic star clusters \\citep[e.g. Westerlund 2 and NGC 3603][]{drew18, drew19}.\nThe fact that WR 3 is located $\\sim$ 100 pc away from a nearby HII-region cluster (see Fig.~\\ref{fig:stellar_census} and~\\ref{fig:sii_oiii}) could support the latter formation scenario.\nIn order to investigate this hypothesis, we have computed the minimum velocity needed for the star to be ejected from the nearby cluster. We assume an age range of (4 $\\pm$ 1) Myr for the WR star\\footnote{We note that, despite the spectral information available, the age also depends on the mass of the progenitor O stars and on rotation effects.}.\nAssuming that the star has been ejected immediately after birth, we find $v_{proj, min} = 25_{-8}^{+5}$ km\/s, which meets the standard runaway criterium of $v > 25$ km\/s \\citep[e.g.][]{drew18, portegieszwart00}.\n\n\\subsection{YSCs age, mass, and ionising flux distribution}\n\\label{section:ysc_properties}\nIn this subsection, we investigate the physical properties of the YSCs, namely their age, mass, and ionising photon flux. The latter, $Q(H^0)$ [s$^{-1}$], is a measure of the number of Lyman continuum (LyC) hydrogen-ionising photons ($h\\nu > 13.6$ eV) emitted per unit time.\n\nFor this purpose, we use the stochastic stellar population synthesis code \\textsc{slug} \\citep[][v2]{dasilva12,krumholz15a}. \n\\textsc{Slug} implements a Monte Carlo technique by randomly drawing stars from an input initial mass function (IMF). This approach allows one to account for the effect of stochastic sampling of the IMF, which is especially important when populating low mass clusters \\citep[see e.g.][]{hannon19}.\nThis is of a particular relevance in our target: NGC 7793 has a very low SFR $\\sim$ 0.1 $M_\\sun$ yr$^{-1}$ \\citep{calzetti15}, resulting in \\ion{H}{ii} regions hosting low mass ($\\sim$ a few 1000s $M_\\sun$) clusters.\nThe \\textsc{SLUG} software package includes the \\texttt{cluster\\_slug} tool~\\citep{krumholz15a}, that implements Bayesian inference techniques to calculate posterior probability distribution functions (PDFs) of physical parameters of star clusters based on  observed cluster photometry, as described in \\citet{krumholz15b}.\nThis is achieved by comparing the data to a large library of simulated clusters. We modified \\texttt{cluster\\_slug} to also return the ionising photon luminosity $Q(H^0)$ as a further output parameter and ran the code on HST broadband photometry of the YSCs. We consider age $t$, mass $M$ and visual extinction $A_V$ as free parameters, and assume a flat prior in $A_V$ and in $\\log t$, and a $\\log(M) \\sim 1\/M$ prior on the mass. We use the library of mock star clusters described in~\\citet{ashworth18}, a Milky Way extinction law by \\citet{fitzpatrick99} and the non-rotating solar metallicity stellar population models from \\citet{genevamodels12}. The abundance map derived in Sect.~\\ref{section:metallicity} confirms that NGC 7793 has a metallicity close to solar \\citep[see also][]{pilyugin14}; the same metallicity was also assumed in the \\textsc{slug} models of \\citet{krumholz15b}. We discuss the impact of including binary stars and the effect of stellar rotation on the resulting physical properties in Sect.~\\ref{section:discussion}.\n\n\\par Figure~\\ref{fig:ysc_distr_FOV} shows the spatial distribution across the FoV of YSC ages (left panel) and masses (right panel). We are showing here as best estimate the median of each cluster PDF.\nAs remarked in \\citet{krumholz15b}, reducing a full PDF to a single point inevitably leads to some imprecision, especially if the PDF is multi-peaked. We follow the method tested in \\citet{krumholz15b} and we refer the interested reader to this work for evaluations of the methodology and effects on the recovered cluster physical parameters.\nIn Fig.~\\ref{fig:ysc_agemass_hist}, we show the age, mass, and ionising flux PDF for YSCs located inside \\ion{H}{ii} regions (in blue) and for field clusters (in orange). Thin and thick lines in the plot indicate, respectively, the PDFs of single clusters and the combined PDFs obtained by summing the fractional probability of each cluster in every age, mass, and flux bin, and re-normalising the resulting distribution.\nIn the case of the age, both the single and combined PDFs are (most often) multi-peaked; the main peak of the combined PDF\nis located at $t \\sim 5$~Myr, both for field and \\ion{H}{ii} region clusters. However, we see that the PDF of field clusters flattens towards older ages, while the probability distribution of \\ion{H}{ii} region clusters is more peaked at ages below 10 Myr, that is clusters located in \\ion{H}{ii} regions have a higher probability to be young and potentially ionising sources.\nThe mass distribution is single-peaked for both categories of clusters, with a tail towards higher masses. The peak of the distribution is located at $\\log(M) = 3.0$ and 3.4~$M_\\odot$ for \\ion{H}{ii} region clusters and field clusters respectively.\nThe $Q(H^0)$ distribution is, as the age, multi-peaked (\\text{both the single and combined PDFs}) with two main modes (both for \\ion{H}{ii} region and field clusters) at $\\log(Q(H^0)) = 44.1$ and 48.8 s$^{-1}$.\nNevertheless, we notice that clusters in \\ion{H}{ii} regions have a higher probability to be associated with the high $Q(H^0)$ value, while the opposite is true for field clusters. In general however, the inferred masses and $Q(H^0)$ are on the low range for YSCs: for a reference, the peak at $\\log(Q(H^0)) = 48.8$ s$^{-1}$ corresponds to the emission of a single O7.5 star \\citep{martins05}. We comment more on the mass distribution below (Fig.~\\ref{fig:mass_distr_gmc_ysc} and associated text).\n\nIn synthesis, we observe that YSCs in \\ion{H}{ii} regions have a higher probability to be younger, slightly lower mass, but a more significant source of ionising photons than clusters in the field. \nThese trend can easily be visualised in the plots of Fig.~\\ref{fig:ysc_distr_FOV}, where we colour code the positions of the clusters in the FoV according to their median ages and masses.\n\n\\par The age range observed for the clusters within the \\ion{H}{ii} regions is in agreement with studies of YSCs in local galaxies \\citep{whitmore11, hollyhead15, hannon19, grasha19}, that find that clusters emerge from their natal gas in $< 4 - 5$ Myr, but that the process can start as early as $2 - 3$ Myr. Also simulations of cluster formation including solely the effect of photoionisation and stellar winds \\citep{dale14, bending20}, have shown that within 3 Myr, and before the onset of SN, several clusters have vacated some channels with low gas density. These channels are easily ionised; therefore ionising radiation can escape from the \\ion{H}{ii} regions even if star formation is still taking place. In Fig.~\\ref{fig:stellar_census} we observe that a significant amount of molecular gas is still detectable in the \\ion{H}{ii} regions hosting YSCs, in agreement with the findings of \\citet{dale14}.\n\n\\par Finally, in Fig.~\\ref{fig:mass_distr_gmc_ysc} we show the total demographics of GMCs (in green), YSCs with age $< 10$ Myr (in blue) and \\ion{H}{ii} regions (in black) within our FoV, highlighting the various phases of the star formation cycle. We observe that GMCs, hosting the molecular gas that will give rise to the next generation of stars, span a rather low range of masses compared to what observed in other local galaxies \\citep[$\\log M_{GMC} \\sim 4.5 - 8 M_\\odot$,][]{hughes16}.\n\nThe collapse of the clouds results in turn in low-mass YSCs. While there is not a one-to-one correlation between the mass spectrum of GMCs and YSCs, we observe that overall the two distributions would suggest an integrated SFE of a few percent of what is typically observed in local galaxies \\citep[see e.g. the review by][]{krumholz19}.\nThe bulk of the YSCs have masses around 1000 $M_\\odot$, and the distribution reaches up to $\\sim 10^{3.5}~M_\\odot$. These mass distributions are typical of galaxies with low SFR, such as M31 \\citep{larsen09, johnson17, adamo20}.\nIf we consider that the ionising feedback from a cluster of 1000 $M_\\odot$ gives rise to an H$\\alpha$ luminosity\nof $\\log(L) \\sim 37.3$ erg s$^{-1}$,\\footnote{\\textsc{Starburst99} \\citep{leitherer99} estimate assuming a \\citet{kroupa01} IMF in the stellar mass range $0.1-120~M_\\odot$.}\nthen the observed luminosity distribution is consistent with the mass distributions of the YSCs in the FoV. If compared to studies of \\ion{H}{ii} regions luminosity functions in the local universe, our sample occupies the low-L end of the luminosity function of local spirals \\citep[$\\log(L ) \\sim 37 - 40$ erg s$^{-1}$,][]{kennicutt89a}. Overall, the low masses and luminosity observed are in agreement with the flocculent morphology and low SFR of our target.\n\n\\begin{figure*\n\\includegraphics[width=8.3cm]{E_BV_map_leguscontour.pdf}\n\\includegraphics[width=9.6cm]{E_BV_map_24microncontour.pdf}\n\\caption{Extinction map derived from the ionised gas observed with MUSE assuming an intrinsic H$\\alpha$\/H$\\beta$ ratio (from Paper~\\textsc{i}) compared to: \\textit{Left panel}: contours of the extinction map derived by~\\citet{kahre18} from the individual stellar extinctions obtained from HST data. The contours correspond to values of $E(B-V) = 0.2 - 0.5$, in steps of 0.033. \\textit{Right panel}: 24~$\\mu$m emission from Spitzer\/MIPS. The contours correspond to a flux of [1.5, 2, 3, 4, 6, 9, 11] MJy\/sr. The thick orange contours indicate the position of GMCs.}\n\\label{fig:extinction_map}\n\\end{figure*}\n\n\\section{Comparison between stellar and ionised gas extinction with maps of cold dense gas and hot dust emission}\n\\label{section:extinction}\nThe HST, MUSE, ALMA, and Spitzer\/MIPS\\footnote{Publicly available in the NED archive \\url{http:\/\/ned.ipac.caltech.edu}.} coverage gives us the rare opportunity of comparing extinctions as traced by stellar reddening and the ionised gas phase to the distribution of the cold dense gas and the hot dust emission.\nIn Paper~\\textsc{i}, we have derived an extinction map from the MUSE H$\\alpha$\/H$\\beta$ ratio using \\textsc{pyneb} \\citep{luridiana15}. Hereby, we assumed a theoretical H$\\alpha$\/H$\\beta_{int}$ = 2.863 and we obtained the observed ratio by binning the stellar continuum subtracted cube with the Voronoi technique of \\citet{cappellari03} to a S\/N = 20 in H$\\beta$ and fitting both lines with a single Gaussian profile.\nHere, we compare the gas extinction map derived in Paper~\\textsc{i} with the map derived by~\\citet{kahre18} from the individual stellar extinctions obtained from HST, and correlate the two maps with the position of GMCs traced by ALMA.\nFigure~\\ref{fig:extinction_map} (left panel) shows overlaid on the gas extinction map the contours of the weighted average map with adaptive resolution from~\\citet{kahre18}. The latter was derived by spatially binning the HST data in the smallest possible regions (of varying size between 1 and 10 arcsec$^2$) hosting at least 10 O stars. In order to compare the two maps, we have converted the colour excess E(V$-$I) to E(B$-$V) assuming a Milky Way extinction law with $R_V = 3.1$, as in \\citet{kahre18}.\nWe observe a clear correspondence between the two maps, despite the difference in binning and in the method used to derive the extinction. In general, regions in which we measure a high extinction also feature a comparable stellar extinction, although the position of the extinction peaks is slightly shifted in the two maps. Conversely, the ionised gas extinction seems to miss some of the regions of enhanced stellar extinction. We notice that these regions lie outside the bright ionised gas, where H$\\beta$ is poorly detected and the ionised gas might not be a sensitive enough tracer of extinction.\nHowever, we observe that the gas extinction seems to better correlate with the position of the GMCs (orange contours in Fig.~\\ref{fig:extinction_map}). We also note that there seems to be a better agreement between gas extinction peaks and GMC position within \\ion{H}{ii} regions.\n\\par\nIn the right panel of Fig.~\\ref{fig:extinction_map}, we furthermore show overlaid on the gas extinction map contours of 24~$\\mu$m emission from Spitzer\/MIPS. The 24~$\\mu$m emission is tracing hot dust in the galaxy, and inside the \\ion{H}{ii} regions it is indicative of where star formation is taking place. We observe a good correlation between gas extinction, GMC position and 24~$\\mu$m emission, indicative of the fact that, in HII regions, the dust is well mixed with gas. We comment more on this fact in relation with the effect of removal of LyC photons by dust prior to their absorption by neutral hydrogen in Sect.~\\ref{section:discussion}.\n\n\\begin{figure*\n\\includegraphics[width=9.2cm]{logO_H_map_1.pdf}\n\\includegraphics[width=9.0cm]{logO_H_map_2.pdf}\n\\caption{Oxygen abundance for the subset of \\ion{H}{ii} regions labelled in Fig.~\\ref{fig:sii_oiii}, determined with the $S$-strong line calibration of \\citet{pilyugin16}. The location of YSCs is indicated with filled circles, colour-coded based on their age (best estimates derived from the median of each PDF). Red crosses and purple stars correspond to the position of WR stars and PNe, respectively. Dark red contours indicate a ratio of [\\ion{S}{ii}]\/[\\ion{O}{iii}] = 0.5 (see Fig.~\\ref{fig:sii_oiii}).}\n\\label{fig:logO_H_map}\n\\end{figure*}\n\n\\section{Mapping the Oxygen abundance within the \\ion{H}{ii} regions}\n\\label{section:metallicity}\nWe estimate an oxygen abundance in the \\ion{H}{ii} regions using the strong-line method of \\citet{pilyugin16}.\nIn Paper~\\textsc{i}, we derived electron temperature and density from the [\\ion{S}{iii}]6312\/9069 and [\\ion{S}{ii}]6716\/6731 line ratios\\footnote{As remarked in Paper~\\textsc{i}, the [NII]5755 `auroral' line used for temperature diagnostics falls  in the wavelength range blocked out in the MUSE\/AO mode.}; however, due to the weakness of the [\\ion{S}{iii}]6312 line and to the fact that the [\\ion{S}{ii}] ratio is largely insensitive to $n_e < 30$ cm$^{-3}$, we were only able to obtain a coarse $T_e$ map and marginally constrained $n_e$, preventing us from measuring direct abundances.\nThe \\citet{pilyugin16} method bases on ratios of strong lines calibrated on a sample of $\\sim$ 300 \\ion{H}{ii} regions. Here we make use of the $S$ calibration, based on the following three line ratios:\n$$N_2 = (I_{[\\ion{N}{ii}]}\\lambda6548 + I_{[\\ion{N}{ii}]}\\lambda6584) \/ I_{H\\beta},$$\n$$S_2 = (I_{[\\ion{S}{ii}]}\\lambda6716 + I_{[\\ion{S}{ii}]}\\lambda6731) \/ I_{H\\beta},$$\n$$R_3 = (I_{[\\ion{O}{iii}]}\\lambda4959 + I_{[\\ion{O}{iii}]}\\lambda5007) \/ I_{H\\beta}.$$\nWe use the upper branch of the calibration, valid for $\\log N_2 \\gtrsim -0.6$, as our data lie well above this limit throughout the FoV. The line ratios are determined from the reddening corrected fluxes, which have been obtained by binning the data with the Voronoi technique to a S\/N = 20 in H$\\beta$ as described in Sect.~\\ref{section:extinction}.\nFigure~\\ref{fig:logO_H_map} shows the resulting metallicity variation in a subset of the \\ion{H}{ii} region sample, for which we construct a detailed photoionisation budget in Sect.~\\ref{section:budget}. The exact location of these regions in the FoV is shown in Fig.~\\ref{fig:sii_oiii}. We observe that the abundance ranges from $12 + \\log(O\/H) \\sim 8.25$ to 8.50, with a median value of 8.37.\nThis range of values is in agreement with the strong-line estimate of \\citet{pilyugin14} for NGC 7793 and with the metallicity range inferred in Paper~\\textsc{i} when comparing our data with the models of star forming galaxies from \\citet{levesque10} in `BPT' diagrams \\citep{baldwin81}.\n\n\\par In small and isolated \\ion{H}{ii} regions, such as regions 4, 5, 6, and 7, we notice that the oxygen abundance varies of about 0.1 dex at most. The largest \\ion{H}{ii} region complexes, such as regions 1, 2, 3, and 8 show larger gradients and pockets of enriched gas coincident or arranged in shell-like structures around clusters and massive stars\\footnote{We omit to plot the position of O stars, which are also a source of enrichment, to avoid overcrowding and facilitate the visualisation of the abundance variation. We refer the reader to Fig.~\\ref{fig:stellar_census}.}.\nThis is particularly visible in region 4, where the YSC is surrounded by an enriched shell, and in regions 6 and 8 (centre left and bottom right) where we estimate a local enhancement in the oxygen abundance at the location of YSCs.\nWe furthermore observe that some of the WR stars are associated with enriched shells or pocket of ionised gas: this is the case in region 1 (WR2) and 8 (WR4), likely due to nitrogen-enriched gas expelled by their stellar winds. In other cases, we do not observe any enrichment surrounding the stars: this is especially visible in region 3 and in the north-west, and south of region 1.\nFinally, we observe a large enhancement of the abundance at the location of one of the planetary nebulae identified in Paper~\\textsc{i} (PN2).\n\n\\par We would like to point out that, although such indirect methods are generally calibrated for unresolved \\ion{H}{ii} regions, they have recently been applied also to study abundance variations within resolved \\ion{H}{ii} regions \\citep[e.g.][]{james16, mcleod19}.\nUsing narrow-band HST maps, \\citet{james16} show chemical inhomogeneity ($\\sim$ 0.1 dex) on scales smaller than 50 pc within the star forming region Mrk 71. \\citet{mcleod19} also report abundance variations and structures similar to what we observe here in two star-forming regions of the Large Magellanic Cloud, resolved at scales of a few parsec. \nHowever, as remarked, for example, by \\citet{mcleod19}, abundances derived with a strong line method can have a secondary non-trivial dependence on temperature and density, resulting for instance in falsely low abundances in regions with a high ionisation state~\\citep{ercolano12,mcleod19,mcleod16a}. In Fig. \\ref{fig:logO_H_map}, we show contours of [\\ion{S}{ii}]\/[\\ion{O}{iii}] = 0.5 (dark red), delimiting strongly ionised regions in the nebulae (see Sect.~\\ref{section:optical_depth}). We observe indeed a close correspondence between the contours and areas of low metallicity in all the regions, pointing to the fact that a degeneracy between oxygen abundance and ionisation conditions is likely playing a role closest to the young massive stars. \nIn the future, IFU having a spectral and spatial capability comparable to MUSE, but covering a bluer wavelength range \\citep[e.g. BlueMUSE,][]{bluemuse}\nwill help us to better understand abundances in resolved \\ion{H}{ii} regions by addressing all such degeneracies.\n\n\\begin{figure\n\\center\n\\includegraphics[width=8cm]{cloudy_model_2Myr.pdf}\n\\caption{\\textsc{cloudy} model illustrating the ionisation structure of an ideal ionisation-bound nebula surrounding a 2 Myr YSC of $M = 10^4~M_\\odot$ at the metallicity of NGC 7793. The radial variation in ionic fraction of \\ion{H}{i} (dashed black), \\ion{O}{iii} (dashed purple), \\ion{S}{ii} (solid green) and the \\ion{S}{ii}\/\\ion{O}{iii} ratio (solid grey) are shown. The nebula consists of two main zones: an inner zone dominated by \\ion{S}{iii} and \\ion{O}{iii}, and an outer zone dominated by \\ion{S}{ii} and \\ion{O}{ii}.}\n\\label{fig:cloudy_model}\n\\end{figure}\n\n\\begin{figure*\n\\center\n\\includegraphics[width=14cm]{s2o3_selected_regions.pdf}\n\\caption{[\\ion{S}{ii}]6716,31\/[\\ion{O}{iii}]4959,5007 ratio map (reddening corrected), tracing the ionisation structure of the gas.\nThe cyan contours indicate a ratio of 0.5, which we use as fiducial limit for the transition to optically thick gas.\nNumbered black contours correspond to the \\ion{H}{ii} regions analysed in Sect~\\ref{section:budget}. The position of WR stars (yellow crosses) and planetary nebulae (white plus symbols) is also indicated.}\n\\label{fig:sii_oiii}\n\\end{figure*}\n\n\\begin{figure\n\\center\n\\includegraphics[width=7.8cm]{hii_class.pdf}\n\\caption{Values of [\\ion{S}{ii}]6716,31\/[\\ion{O}{iii}]4959,5007 along the contour of the \\ion{H}{ii} regions labelled in Fig.~\\ref{fig:sii_oiii}. The solid points and error bars show, respectively, the median and the range in values spanned by the ratio. The cyan line indicates a ratio of 0.5, which we use as fiducial limit for the transition to optically thick gas. Regions for which the range in [\\ion{S}{ii}]\/[\\ion{O}{iii}] extends below this limit are classified as featuring optically thin channels (CH); all other regions are classified as ionisation bounded (IB).}\n\\label{fig:hii_class}\n\\end{figure}\n\n\\begin{figure\n\\center\n\\includegraphics[width=9cm]{fesc_vs_appearance.pdf}\n\\caption{Ratio of expected to observed ionising luminosity $Q(H^0)$ for the regions labelled in Fig.~\\ref{fig:sii_oiii} versus their visual appearance: ionisation bounded regions (IB, green stars) vs regions featuring optically thin channels (CH,  blue stars). The dashed horizontal line indicates an $f_{esc} = 0$.}\n\\label{fig:fesc_vs_appearance}\n\\end{figure}\n\n\\section{Ionisation structure of the \\ion{H}{ii} regions}\n\\label{section:optical_depth}\n\nThe ionisation structure of a nebula can be studied through the relative emission of two ions with different ionisation potential. The structure of an ideal ionisation bounded \\ion{H}{ii} region is shown in Fig.~\\ref{fig:cloudy_model}.\nThe figure illustrates a model generated with \\textsc{cloudy} for a nebula at the metallicity of our target ($Z \\sim 0.008$, see Sect.~\\ref{section:metallicity}), surrounding a 2 Myr YSC with a mass of $10^4~M_\\odot$. The simulation is run until the gas interface reaches an optical depth $\\tau = 2$.\nTwo zones can be distinguished in the nebula: a central zone with a higher ionisation state, dominated by S++ and O++ and an external zone with a lower ionisation state, where these ions are singly ionised. The ratio of S+\/O++ (thick grey) can then be used as a proxy for the optical depth of the nebula, that progresses from optically thin in the inner zone to optically thick in the outer zone, with a sharp cutoff at the Str\u00f6mgren radius.\nIn reality, the inner zone is often not, or only partially embedded in the optically thick outer envelope, so that a fraction of hydrogen-ionising photons are not absorbed and can escape into the ISM.\nStudying the spatial extent of the two zones through emission from their most abundant ions can provide us with information about the ionisation structure of an \\ion{H}{ii} region. \\citet{pellegrini12} exploited this dependency to develop the ionisation parameter mapping (IPM) method and applied it to study the optical depth of \\ion{H}{ii} regions in the Large and Small Magellanic Clouds (LMC and SMC) with narrow-band photometry in [\\ion{S}{ii}], [\\ion{O}{iii}] and H$\\alpha$. Regions were classified based on the fraction of the central high-ionisation region being surrounded by a low-ionisation envelope in an [\\ion{S}{ii}]\/[\\ion{O}{iii}] emission map. Regions completely embedded in the envelope are labelled as optically thick, regions with a low [\\ion{S}{ii}]\/[\\ion{O}{iii}] throughout as optically thin and those with an in between morphology as `blister nebulae'. \nEach morphological class was then assigned an escape fraction; we discuss the results of this approach in the light of our work in Sect.~\\ref{section:discussion}.\n\n\\par Figure~\\ref{fig:sii_oiii} shows a map of the [\\ion{S}{ii}]6716,31\/[\\ion{O}{iii}]4959,5007 ratio in our FoV. The map has been reddening corrected assuming an intrinsic H$\\alpha$\/H$\\beta$ ratio of 2.863, as described in Paper~\\textsc{i}. The data have been Voronoi binned to a S\/N = 20 in H$\\beta$. The numbered black contours in Fig.~\\ref{fig:sii_oiii} indicate the subset of the \\ion{H}{ii} regions - out of the full sample identified in Paper~\\textsc{i} and shown in Fig.~\\ref{fig:stellar_census} - that we study in detail in Sect.~\\ref{section:budget}.\nThe bar shows increasing ratios of [\\ion{S}{ii}]\/[\\ion{O}{iii}], from black to yellow, corresponding to an increasing optical depth of the medium (or, a decrease in its transparency to ionising photons). A ratio of 1 (purple) indicates where the abundance of S+ prevails over O++, as shown in the model in Fig.~\\ref{fig:cloudy_model}\\footnote{We caution however that Fig.~\\ref{fig:cloudy_model} shows the ratio of the ionic fractions, which does not compare directly to a line emissivity ratio.}.\n\n\\par We observe that the DIG has a lower ionisation state than the \\ion{H}{ii} regions (higher ratios of [\\ion{S}{ii}]\/[\\ion{O}{iii}]),\nas we also observed in Paper~\\textsc{i} by analysing the ratios of [\\ion{S}{ii}] and [\\ion{N}{ii}] to H$\\alpha$ and in agreement with measurements in nearby galaxies \\citep{haffner09}.\nIn about half of the regions we observe zones of optically thin gas extending until the regions edge, which could potentially indicate escape channels cleared by stellar feedback. In Sect.~\\ref{section:budget} we explore the link between the fraction of photons escaping the regions and the ionisation structure traced by [\\ion{S}{ii}]\/[\\ion{O}{iii}].\n\nIn Table~\\ref{table:obs_properties_regions} we summarise the visual appearance of the regions in [\\ion{S}{ii}]\/[\\ion{O}{iii}], with a similar approach as \\citet{pellegrini12}. Regions are classified\ninto a `classical' ionisation bounded nebula (`IB'), with an ionisation structure as the model in Fig.~\\ref{fig:cloudy_model}, or a nebula featuring one (or more) optically thin channels (`CH') cutting through the low ionisation zone and reaching the regions edge.\nWe base our classification on the value of [\\ion{S}{ii}]\/[\\ion{O}{iii}] along the region contour, as shown in Fig.~\\ref{fig:hii_class}. In the Figure, the solid points indicate the median value of the ratio along the contour, and the grey bars span the entire range of observed values. In order to determine which regions provide a potential escape path, we consider [\\ion{S}{ii}]\/[\\ion{O}{iii}] = 0.5 as fiducial limit for the transition to optically thick gas: regions having a ratio below this threshold along their contours are classified as CH; all other regions are classified as IB.\nWe test the effect of spatial binning on this classification by using both a finer and a coarser binning pattern, corresponding to S\/N = 15 and 25 in H$\\beta$. The classification remains unchanged, suggesting that spatial tessellation does not play a decisive role.\nWe stress that this classification does not take into account the geometry of the nebula and can only be conclusive when having a statistically significant sample of regions that allows to eliminate selection effects.\n\n\\begin{table*}\n\\centering\n\\caption{Spectral classification of WR stars.}\n\\begin{tabular}{ccccc}\n\\hline \\hline\nObj. ID & Ra (J2000) & Dec (J2000) & Features\/Reference & Classification \\\\ \\hline\nWR1 & 23:57:40.81 & -32:35:36.5 & (1) & WN2 \\\\\nWR2 & 23:57:41.43 & -32:35:35.9 & (1) & WC4 \\\\  \nWR3 & 23:57:45.74 &-32:34:37.4 & Nebular lines + \\ion{He}{ii} 4686 are weak, H$\\beta$ absorption & WN + O star \\\\\nWR4 & 23:57:44.75 &-32:34:25.1 & \\ion{N}{iii} 4640, \\ion{He}{ii} 4886 & WN (late-type) \\\\\nWR5 & 23:57:40.90 &-32:35:35.6 & \\ion{He}{ii} 6560 (strong and broad) & WC4\\\\\nWR6 & 23:57:40.53 & -32:35:33.1 & \\ion{N}{iii} 4640, \\ion{He}{ii} 4686 & WN (late-type) \\\\\nWR7 & 23:57:41.37 &-32:35:52.5 & \\ion{N}{v} 4603 + \\ion{N}{iii} 4640 (in equal strength) & WN (mid-type) \\\\\nWR8 & 23:57:43.27 & -32:35:49.6 & \\ion{He}{ii} 4686 (narrow), \\ion{N}{iii} 4640 & WN (late-type)\\\\\nWR9 & 23:57:40.94 &-32:34:47.3  & Analogue to WR8, slightly broader \\ion{He}{ii} & WN (late-type)\\\\\n\\hline\n\\end{tabular}\n\\tablebib{(1) \\citet{bibby10}}\n\\label{table:wr_stars}\n\\end{table*}\n\n\\section{Ionisation budget}\n\\label{section:budget}\n\nWe construct a ionisation budget for the entire FoV and inspect the \\ion{H}{ii} regions labelled in Fig.~\\ref{fig:sii_oiii} in detail.\nThe latter regions were selected to lie for their most part in the FoV, and not at the edge of the field.\nWe note that we consider here the outermost contours of the \\ion{H}{ii} regions sample, based on a flux brightness cut in H$\\alpha$ = $\\SI{6.7e-18}{\\erg \\per \\second \\per \\centi \\metre \\squared}$spaxel$^{-1}$ as defined in Paper~\\textsc{i}. We therefore consider the largest \\ion{H}{ii} region complexes (such as regions 1 and 8 in Fig.~\\ref{fig:sii_oiii}) in their entirety. This choice is made because dividing the regions within smaller subregions would imply arbitrary choices of their outermost border and division of their fluxes.\n\n\\par For each region, we model the expected $Q(H^0)$ from the stellar content as described below, and compare it to the observed ionising flux computed from the (reddening corrected) H$\\alpha$ luminosity, following the calibration of \\citet{kennicutt98}:\n$$ Q(H^0)_{obs} = 7.31 \\times 10^{11} L(H\\alpha) \\quad [\\mbox{erg s}^{-1}],$$\nbased on the assumption of case B recombination and an electron temperature $T_e \\sim \\num{10 000}$ K.\n\n\n\\subsection{O stars}\nDue to the selection method for the O stars catalogue (see Sect.~\\ref{section:data}), we lack information such as mass, age or spectral class of the O stars. In order to compute what $Q(H^0)$ is expected from the O stars population, we therefore have to assume a mass distribution. We consider a Salpeter mass function \\citep{salpeter55}:\n$$ p(m) = A m^{-\\alpha},$$\nwhere $p(m)$ denotes the probability of finding a star of mass $m$, and $\\alpha = 2.35$. We sample the distribution from $m_{min} = 20 M_\\odot$, the lowest stellar mass probed by the catalogue, to $m_{max} = 60 M_\\odot$ with 1000 Montecarlo realisations.\nThis upper limit corresponds to bright HII regions in the Milky Way \\citep{feigelson13} and is the highest stellar mass tabulated in the work of \\citet{martins05} (see below). We also obtain a lower limit for $Q(H^0)$ of field stars by sampling the Salpeter distribution up to a maximum mass  $m_{max} = 30~M_\\odot$, corresponding to the most massive stars observed in typical galactic star forming regions \\citep{bik10, gennaro12, bik12, ellerbroek13}. Moreover, stars of mass $M > 20~M_\\odot$ are generally expected to emit enough ionising radiation to be embedded in an \\ion{H}{ii} region, and if observed in the field are most likely runaway objects.\nFor each Montecarlo realisation, we compute the total number of O stars by taking into account the 30\\% contamination rate and $\\sim$ 70\\% completeness of the stellar catalogue (see Sect.~\\ref{section:data}):\n$$n_{Ostars,tot} = \\mathcal{N}(n_{Ostars}, 0.3),$$\nwhere $\\mathcal{N}$ denotes a normal distribution. Hereby, we also excluded O stars coinciding with the position of WR stars and YSCs (r $<$ 0.4'').\nWe then draw $n_{Ostars,tot}$ times a mass\n$$m_i = p(m < m') = \\int_{m_{min}}^{m'} p(m) dm = m_{min} \\left[C \\cdot \\mbox{unif}(0,1) + 1 \\right]^{1\/(1 - \\alpha)}$$\n$$ C = \\left( \\frac{m_{max}}{m_{min}} \\right)^{1 - \\alpha} - 1,$$\nwhere \\texttt{unif} denotes a uniform distribution. The resulting masses are then converted into $Q(H^0)$ values by interpolation of Table 1 from \\citet{martins05}, and the total flux of the region $Q(H^0)_{tot}$ is stored for each iteration.\nWe finally compute the best $Q(H^0)$ value and its uncertainty from, respectively, the median $m$ and $m \\pm 1 \\sigma$ of the resulting $Q(H^0)_{tot}$ distribution.\n\n\\subsection{YSC}\nWe determine $Q(H^0)$ for the YSC using the \\textsc{slug} code, as described in Sect.~\\ref{section:ysc_properties}.\nWe obtain a distribution of $Q(H^0)_{tot}$ from 1000 Montecarlo realisations: for each realisation, we sample one value from the PDF of each cluster in the region, and sum the resulting quantities. As for the O stars, we then compute the best $Q(H^0)$ value and its uncertainty from, respectively, the median $m$ and $m \\pm 1 \\sigma$ of the resulting $Q(H^0)_{tot}$ distribution.\nHereby, we also check that no YSCs are coincident with the position of WR stars, within a radius of 0.4''.\n\n\\subsection{WR stars}\nLastly, we compute the contribution of WR stars. We classify each star as Carbon-dominated (WC type) or \\ion{He}{ii}-dominated (WN-type), based on its spectral features as listed in Table~\\ref{table:wr_stars}. We then compute $Q(H^0)_{exp}$ from Tables 3 and 4 in \\citet{smith02}, assuming a temperature $T = \\num{60000}$ K for WN-type stars and $T = \\num{120000}$ K for WC-type stars. We compute the uncertainty on $Q(H^0)$ with 1000 Montecarlo realisations of $T$ in the range $[40 - 80] \\times 10^3$ K for WN-type stars and $[100 - 140] \\times 10^3$ K for WC-type. Also in this case, we consider the median $m$ and $m \\pm 1 \\sigma$ of the resulting $Q(H^0)_{tot}$ distribution.\n\n\\begin{table*}\n\\centering\n\\caption{Observed properties of the sub-sample of regions shown in Fig.~\\ref{fig:sii_oiii} and of the overall \\ion{H}{ii} regions and DIG population. We note that the number of O stars indicated here is not completeness-corrected. $t_{max}$ indicates the age of the oldest cluster in each region (fiducial value of 50 $\\pm$ 25 percentile of the age PDF; masking cluster with ages $> 10$ Myr). In the case of region 5, which is hosting exclusively O stars, we have assumed 3 Myr as an upper limit for the age. The indicated E(B-V) is the median extinction in the region (estimated from the Balmer decrement).\nThe total $L(H\\alpha)$ in each region is reddening corrected. Regions are classified as described in Sect.~\\ref{section:optical_depth} into `classical' ionisation bounded regions (IB) and regions featuring optically thin channels (CH).}\n\\begin{tabular}{lcccccccc}\n\\hline \\hline\nRegion ID &\n\\multicolumn{3}{c}{Stellar content} &\n$t_{max}$ &\n$\\log M_{GMC}$ & E(B-V) &\n$\\log L(H\\alpha$) & Classification \\\\\n & \\small{YSC} & \\small{O$\\star$} & \\small{WR} & [Myr] & $[M_\\odot]$ & & $[\\mbox{erg s}^{-1}]$\\\\\n\\hline\n\\textbf{1} & 24 & 111 & 6 & $5.7_{-1.8}^{+1.8}$ & 6.1 & 0.14 & 39.11 & CH\\\\[0.1cm]\n\\textbf{2} & 3  & 30  & 1 & $4.3_{-2.9}^{+2.6}$ & 6.0 & 0.13 & 38.68 & CH\\\\[0.1cm]\n\\textbf{3} & 4  & 26  & 1 & $3.9_{-2.5}^{+2.9}$ & 5.3 & 0.19 & 38.73 & CH\\\\[0.1cm]\n\\textbf{4} & 1  & 19  & 0 & $5.7_{-1.4}^{+1.2}$ & 4.5 & 0.12 & 38.25 & IB\\\\[0.1cm]\n\\textbf{5} & 0  & 5   & 0 & $\\leq 3.0$ & 3.4 & 0.14 & 37.35 & IB\\\\[0.1cm]\n\\textbf{6} & 1  & 2   & 0 & $1.7_{-1.2}^{+2.2}$ & --- & 0.09 & 37.41 & CH\\\\[0.1cm]\n\\textbf{7} & 1  & 2   & 0 & $3.2_{-2.1}^{+3.6}$ & --- & 0.11 & 37.01 & IB\\\\[0.1cm]\n\\textbf{8} & 8  & 110 & 1 & $4.3_{-2.6}^{+2.6}$ & 6.3 & 0.17 & 38.89 & CH\\\\[0.1cm]\n\\textbf{Tot \\ion{H}{ii}} & 47 & 404 & 8 & & & & \\\\[0.1cm]\n\\textbf{Tot DIG} & 49 & 271 & 1 & & & &\\\\\n\\hline\n\\end{tabular}\n\\label{table:obs_properties_regions}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{Ionisation budget of the regions shown in Fig.~\\ref{fig:sii_oiii} and of the overall \\ion{H}{ii} regions and DIG population. The last row indicates a lower limit for the modelled ionising flux in the DIG, obtained by considering an upper limit $m_{max} = 30 M_\\odot$ for the field candidate O stars (as opposed to $m_{max} = 60 M_\\odot$ assumed elsewhere). All the uncertainties indicated are $\\pm 1 \\sigma$ errors on the $Q(H^0)$ distribution resulting from the Montecarlo sampling. The uncertainties on $\\log Q^0_{obs}$ are all of order $\\leq 0.001$.}\n\\begin{tabular}{lccccccc}\n\\hline \\hline\nRegion ID &\n$\\log Q^0_{exp, Ostars}$ &\n$\\log Q^0_{exp, YSC}$ &\n$\\log Q^0_{exp, WR}$ &\n$\\log Q^0_{exp, tot}$ &\n$\\log Q^0_{obs}$&\n$Q^0_{exp, tot}\/Q^0_{obs}$ &\n$f_{esc}$\n\\\\\n& [s$^{-1}$] & [s$^{-1}$]& [s$^{-1}$]& [s$^{-1}$] & [s$^{-1}$] & & \\\\\n\\hline\n\n\\textbf{1}\n& $51.11_{-0.14}^{+0.14}$\n& $49.60_{-0.34}^{+1.50}$\n& $50.06_{-0.02}^{+0.02}$\n& $51.16_{-0.13}^{+0.17}$\n& $50.74$\n& $2.71_{-0.78}^{+0.84}$\n& $0.63_{-0.15}^{+0.09}$  \\\\[0.2cm]\n\n\\textbf{2}\n& $50.54_{-0.13}^{+0.16}$\n& $48.54_{-0.42}^{+1.59}$\n& $49.40_{-0.09}^{+0.00}$\n& $50.58_{-0.13}^{+0.16}$\n& $50.31$\n& $1.83_{-0.49}^{+0.65}$\n& $0.45_{-0.20}^{+0.14}$  \\\\[0.2cm]\n\n\\textbf{3}\n& $50.48_{-0.15}^{+0.17}$\n& $48.54_{-0.43}^{+1.59}$\n& $49.40_{-0.09}^{+0.00}$\n& $50.52_{-0.15}^{+0.17}$\n& $50.35$\n& $1.50_{-0.47}^{+0.52}$\n& $0.33_{-0.30}^{+0.17}$  \\\\[0.2cm]\n\n\\textbf{4}\n& $50.34_{-0.15}^{+0.18}$\n& $48.53_{-0.43}^{+0.59}$\n& ---\n& $50.34_{-0.15}^{+0.19}$\n& $49.98$\n& $2.31_{-0.78}^{+0.94}$\n& $0.57_{-0.22}^{+0.13}$  \\\\[0.2cm]\n\n\\textbf{5}\n& $49.70_{-0.25}^{+0.30}$\n& ---\n& ---\n& $49.70_{-0.25}^{+0.30}$\n& $49.10$\n& $3.94_{-2.30}^{+2.74}$\n& $0.75_{-0.36}^{+0.10}$  \\\\[0.2cm]\n\n\\textbf{6}\n& $49.07_{-0.35}^{+0.94}$\n& $48.53_{-0.43}^{+0.59}$\n& ---\n& $49.18_{-0.37}^{+0.86}$\n& $49.14$\n& $1.24_{-0.86}^{+1.81}$\n& $0.20_{-1.83}^{+0.48}$  \\\\[0.2cm]\n\n\\textbf{7}\n& $49.09_{-0.34}^{+0.86}$\n& $48.53_{-0.43}^{+0.59}$\n& ---\n& $49.19_{-0.36}^{+0.80}$\n& $48.75$\n& $3.17_{-2.15}^{+4.09}$\n& $0.68_{-0.67}^{+0.18}$  \\\\[0.2cm]\n\n\\textbf{8}\n& $51.10_{-0.13}^{+0.14}$\n& $48.85_{-0.42}^{+2.04}$\n& $49.39_{-0.08}^{+0.01}$\n& $51.11_{-0.13}^{+0.15}$\n& $50.53$\n& $3.85_{-1.15}^{+1.19}$\n& $0.74_{-0.11}^{+0.06}$  \\\\[0.2cm]\n\n\\textbf{Tot \\ion{H}{ii}}\n& $51.68_{-0.13}^{+0.13}$\n& $49.90_{-0.33}^{+1.04}$\n& $50.23_{-0.02}^{+0.02}$\n& $51.70_{-0.13}^{+0.14}$\n& $51.22$\n& $3.06_{-0.84}^{+0.93}$\n& $0.67_{-0.12}^{+0.08}$  \\\\[0.2cm]\n\n\\textbf{Tot DIG}\n& $51.50_{-0.12}^{+0.13}$\n& $49.95_{-0.35}^{+0.91}$\n& $49.40_{-0.09}^{+0.00}$\n& $51.52_{-0.13}^{+0.15}$\n& $50.65$\n& $7.48_{-2.02}^{+2.39}$\n& $0.87_{-0.05}^{+0.03}$\\\\[0.2cm]\n\n\\textbf{Tot DIG}$_{lower \\thinspace limit}$\n& $50.94_{-0.13}^{+0.13}$\n&\n&\n& $50.99_{-0.15}^{+0.20}$\n&\n& $2.30_{-0.64}^{+0.75}$\n& $0.57_{-0.17}^{+0.11}$\\\\\n\\hline\n\\end{tabular}\n\\label{table:photo_budget}\n\\end{table*}\n\n\n\n\\subsection{Resulting budget}\n\\label{section:budget_results}\nTable~\\ref{table:obs_properties_regions} and \\ref{table:photo_budget} summarise, respectively, the observed properties and the photoionisation budget for the entire FoV\nand in detail for the eight regions labelled in Fig.~\\ref{fig:sii_oiii}.\nIn Table~\\ref{table:obs_properties_regions} we list the stellar content, the age of the oldest star or cluster in each region, the total GMC mass, median $E(B - V)$ (estimated from the Balmer decrement) and H$\\alpha$ luminosity (reddening corrected), as well as the morphological classification derived in Sect.~\\ref{section:optical_depth}.\nIn Table~\\ref{table:photo_budget} we indicate the total flux $Q(H^0)_{exp}$ modelled from the stellar content, as well as the relative contribution from O stars, YSC, and WR stars, and the observed ionising photon flux $Q(H^0)_{obs}$ derived from the reddening corrected H$\\alpha$ luminosity. We also list the ratio of expected to observed flux and the corresponding escape fraction $f_{esc} = 1 - Q(H^0)_{obs}\/Q(H^0)_{exp}$.\nWe note that the extremely large uncertainties on regions 5 and 7 are driven by the fact that these regions host (almost) exclusively O stars, for which $Q(H^0)$ is more loosely constrained. We also note that in region 6 the uncertainty ranges to unphysical values $f_{esc} < 0$, indicating that either the models are underestimating the photon flux, or the observed luminosity is being overestimated, possibly due to the reddening correction or to the exact location of the region boundaries.\n \n\\par From Table~\\ref{table:photo_budget}, we see that WR and O stars dominate the contribution to the $Q^0_{exp, tot}$ value of the regions. Overall, we find an $f_{esc, \\ion{H}{ii}} = 0.67_{-0.12}^{+0.08}$ for the entire population of \\ion{H}{ii} regions (black contours in Fig.~\\ref{fig:stellar_census}). We also observe that the stellar population in the DIG produces a more than sufficient amount of ionising photons ($Q(H^0)_{exp} > Q(H^0)_{obs}$), and that the DIG is therefore consistent with being self-ionised, with $f_{esc, DIG} = 0.87_{-0.05}^{+0.03}$. This holds also if considering a maximum mass of $30~M_\\odot$ for field O stars (last row in Table~\\ref{table:photo_budget}), in which case we find $f_{esc, DIG} = 0.57_{-0.17}^{+0.11}$.\nIn our FoV, we observe hence that the sources of ionising photons produce a photon flux that is more than sufficient to explain the emission of the ionised ISM, both within and outside the \\ion{H}{ii} regions.\n\n\\subsection{Escape fraction from individual \\ion{H}{ii} regions}\n\\label{section:fesc_individual}\n\nIn this section, we focus in more detail on individual \\ion{H}{ii} regions. In order to better visualise the link between $f_{esc}$ and the visual appearance of the regions, in Fig.~\\ref{fig:fesc_vs_appearance} we plot the ratio of $Q(H^0)_{exp}\/Q(H^0)_{obs}$, and label each region according to the morphological criteria described in Sect.~\\ref{section:optical_depth} (IB: ionisation bounded, CH: optically thin channels). We do not see any clear trend with visual appearance; we discuss this result in the light of other studies in Sect.~\\ref{section:discussion}.\n\n\\par In Fig.~\\ref{fig:fesc_vs_agerange}, we furthermore investigate the dependence of the $Q(H^0)_{exp}\/Q(H^0)_{obs}$ ratio on the age of the oldest cluster in each region.\nWe consider the latter as a proxy for the age range spanned by the stellar population inside each \\ion{H}{ii} region, assuming that all regions are currently forming stars.\nWe remark that this approximation might not be accurate for regions 6 and 7, in which no GMCs are detected with ALMA and the extinction traced by the ionised gas is very low (see Table~\\ref{table:obs_properties_regions}); however, both regions are still hosting young candidate O stars.\nWhen determining the age of the oldest cluster, we use as best estimate the median of each cluster PDF, and we exclude clusters with $t > 10$ Myr, as such objects are not anymore associated with \\ion{H}{ii} regions, and are most likely line-of-sight objects. For region 5, which is hosting exclusively O stars, we assume an upper limit of 3 Myr for the age. \nWe also colour-code the regions according to their median $E(B - V)$ (left panel) and total GMC mass (right panel), to indicate which regions are more affected by reddening and are still actively forming stars.\nDust can indeed absorb part of the LyC photons before they have the possibility to ionise hydrogen, and re-emit them at longer wavelengths, mimicking a larger $f_{esc}$.\nWe see that for regions 3 and 8, this could in part explain the $f_{esc} > 0$ observed, whereas - given their low dust and GMC content - regions 4 and 5 could be a true leaking region. We comment more on the effect of photons removal by dust in Sect.~\\ref{section:discussion}. In our small sample of regions and due to the large uncertainties, we do not find any trend between $f_{esc}$ and the age of the stellar population in the region. This is discussed\nin more detail in Sect.~\\ref{section:discussion}.\n\n\\begin{figure*\n\\includegraphics[width=8cm]{fesc_tmax_ebv.pdf}\n\\includegraphics[width=8cm]{fesc_tmax_gmc.pdf}\n\\caption{Ratio of expected to observed ionising photon flux $Q(H^0)$ for the regions labelled in Fig.~\\ref{fig:sii_oiii}, versus the age of the oldest cluster in each region (best-value estimate obtained from the median of the age PDF; masking cluster with ages $> 10$ Myr). For region 5, which is hosting exclusively O stars, we assume an upper limit of 3 Myr for the age. The dashed horizontal line indicates an $f_{esc} = 0$.\n\\textit{Left panel}: the points are colour-coded according to the median $E(B - V)$ in each region (Balmer decrement estimate). \\textit{Right panel}: the regions are colour-coded according to their total GMC mass. Empty black circles indicate regions hosting no GMCs.}\n\\label{fig:fesc_vs_agerange}\n\\end{figure*}\n\n\\section{Discussion}\n\\label{section:discussion}\nIn Paper~\\textsc{i}, we found a DIG fraction $f_{DIG, obs} = 0.15$ from the H$\\alpha$ luminosity in our FoV\\footnote{Revised estimate for the H$\\alpha$-selected \\ion{H}{ii} regions sample and corrected for extinction.}.\nThis is in good agreement with the fraction of diffuse gas expected by modelling the stellar and cluster population:\n$$f_{DIG, exp} = \\frac{Q^0_{exp,DIG} }{ Q^0_{exp,DIG}  + Q^0_{exp,HII}} \\gtrsim 0.17.$$\nMoreover, we assessed from our BPT diagram analysis in paper~\\textsc{i} that the bulk of the DIG emission in our FoV was consistent with being photoionised. Here we confirm that the DIG is consistent with being self-ionised by field stars and clusters, with an overabundance of ionising photons $f_{esc} \\sim 0.87_{-0.05}^{+0.03}$; we confirm this trend also when considering our lower-limit estimate.\nThis result is perhaps surprising considering the fact that we observe in Fig.~\\ref{fig:sii_oiii} that the DIG is not optically thin to Lyman continuum emission (high [\\ion{S}{ii}]\/[\\ion{O}{iii}] ratios). However, if the DIG has a clumpy structure, the emission line ratio could be tracing the denser, optically thick clumps; the apparent lack of H$\\alpha$ emission can originate from photons escaping through the dilute, optically thin interclump medium, which through its low density will not contribute significantly to the observed emission line intensities.\nWe also observe that overall the \\ion{H}{ii} regions are leaking ionising photons at a rate $f_{esc} \\sim 0.67_{-0.12}^{+0.08}$, and that seven out of eight have an $f_{esc} \\gtrsim 0.3$.\n\n\\par We note that the uncertainties on $f_{esc}$ are rather large, as we try to account for a variety of effects, including: the contamination rate $\\sim 30\\%$ in the O stars catalogue, the large range in $Q(H^0)$ obtained by Montecarlo sampling O stars of different mass,\nthe range in values spanned by the PDFs of the clusters and the assumption on the temperature of the WR stars.\nThe large uncertainties recovered are consistent with the work of \\citet{niederhofer16}: by varying parameters of synthetic YSCs, the authors conclude that escape fractions derived from broadband photometry data are typically dominated by uncertainties in the spectral types of the stars.\nHowever, even when considering the uncertainties, we find $f_{esc} > 0$ for the DIG overall and for six out of the eight \\ion{H}{ii} regions inspected  (see Table~\\ref{table:photo_budget} and Fig.~\\ref{fig:fesc_vs_appearance}). \n\n\\par An additional factor that is not taken into account in the computation of $f_{esc}$ is the effect of absorption of LyC photons by dust within the \\ion{H}{ii} regions before these photons have the chance of ionising hydrogen atoms. In Sect.~\\ref{section:extinction}, we have compared the extinction map derived from the H$\\alpha$\/H$\\beta$ ratio with 24~$\\mu$m emission from Spitzer\/MIPS (Fig.~\\ref{fig:extinction_map}, right panel), and found a good correspondence, indicative that dust and gas are well mixed \\citep[see e.g.][]{choi20}. The resolution of mid-IR data is currently insufficient to confirm potential dust absorption in proximity of the sources of Lyman continuum photons at the distance of our target, and will only become possible with new generation IR telescopes such as JWST. However, in local \\ion{H}{ii} regions, it is typically observed that a large fraction of LyC photons contributes to hydrogen ionisation. This fraction anti-correlates with metallicity \\citep[e.g.][]{inoue01}, and at solar metallicity, an increasing fraction of Lyman continuum photons are absorbed by dust, instead. In the metallicity range spanned by our target (12 + log(O\/H) $\\sim  8.25 - 8.5$, see Sect.~\\ref{section:metallicity}), \\citet{inoue01} estimate that $\\gtrsim 80\\%$ Lyman continuum photons are absorbed by neutral hydrogen. Therefore, we expect that the effect of dust absorption in the \\ion{H}{ii} regions in our FoV can reduce $Q(H^0)_{obs}$ of up to $\\sim$ 20\\%, which lies well within the given uncertainties.\n\n\\par Other studies targeting \\ion{H}{ii} regions in nearby galaxies at high resolution found similar escape rates.\n\\citet{doran13} have estimated an $f_{esc} > 0.5$ in the 30 Dor star forming region in the LMC, based on a complete spectroscopic census of hot, luminous stars.\nIn a sample of \\ion{H}{ii} regions in the LMC and SMC studied with narrowband photometry, \\citet{pellegrini12} have found $f_{esc} > 0.4$ for all the regions. \\citet{mcleod19} have studied in more detail two of the largest \\ion{H}{ii} region complexes in the LMC with MUSE, finding $f_{esc} > 0.2$ for the complexes and the respective sub-regions.\nFinally, in two regions recently observed with MUSE and HST in the nearby dwarf galaxy NGC 300, \\citet{mcleod20} have estimated an $f_{esc} \\gtrsim 0.3$ .\n\nFollowing the non-negligible escape of Lyman continuum photons from \\ion{H}{ii} regions, several studies of local and nearby galaxies have proven the DIG consistent with being photoionised. For example, \\citet{pellegrini12} have estimated that both in the LMC and SMC the total galactic escape fraction would be sufficient to account for the observed $f_{DIG}$, and that, when considering the additional contribution of field stars, the galaxies could leak a substantial amount of ionising radiation into the circumgalactic medium. Similarly, in a recent MUSE+HST study of the Antennae merger system, \\citet{weilbacher18} have estimated a sufficient fraction of Lyman-continuum leakage in order to explain the amount of DIG observed.\n\n\\par We do not find any correlation between ionising photon leakage in the HII regions and their visual appearance (`classic' ionisation bound regions vs regions featuring optically thin channels).\nThis is somewhat in contrast with the results of \\citet{pellegrini12} that, by comparing the fiducial $f_{esc}$ assigned to \\ion{H}{ii} regions in the LMC and SMC based on their appearance in [\\ion{S}{ii}]\/[\\ion{O}{iii}] (see Sect.~\\ref{section:optical_depth}) with previous $f_{esc}$ estimated based on the stellar population, found a good agreement overall. On the other hand, a systematic trend is not apparent from the MUSE results of \\citet{mcleod19} on the LMC, that used the [\\ion{O}{ii}]7320,7330\/[\\ion{O}{iii}]4959,5007\nratio as optical depth tracer. We interpret this apparent tension as likely being a simple effect of 3D geometry in number limited samples.\n\\par In our small sample of regions, we do not find evidence for a correlation between age range of the stellar population and $f_{esc}$. Increasing the sample size will be essential to investigate any such correlation, that could be expected in light of the results of galactic and cosmological simulations.\n\\citet{kim13a} simulated ionising radiation and supernova feedback in a low-redshift galactic disk, at a high spatial resolution $\\sim 4$ pc. The study indicated that photons emitted at later ages are more likely to escape the star forming region, as the gas surrounding the region is dispersed by ionising radiation and supernova feedback. \nA similar result was obtained in the recent high-resolution cosmological simulations of \\citet{ma20}, that studied $\\sim$ 30 zoom-in simulations of galaxies at $z \\gtrsim 5$.\nThe simulations suggest that ionising photons are preferably leaking from star forming regions containing a kpc-size superbubble, likely created by clustered SNe set off by stars of age $>$ 3 Myr. As the bubble expands, new stars are formed at its edge, inside a dense shell of compressed gas. The shell keeps expanding while forming stars and, as a consequence, the young stars formed in it end up inside the superbubble and are able to fully ionise channels of low-column density  pre-cleared by feedback from the previous population of stars. Therefore, regions hosting stars spanning a large range in age seem to be advantaged in leaking photons. Also observations have already pointed to the importance of the local star formation history in shaping the ionisation conditions in \\ion{H}{ii} regions. By analysing a sample of $\\sim$ 5000 \\ion{H}{ii} regions from CALIFA \\citep[Calar Alto Large Integral Field Area survey,][]{sanchez12}, \\citet{sanchez15} found for example a correlation between the position occupied by the regions in BPT diagrams and the age and metallicity of their stellar population.\n\n\\par Finally, we comment on the impact of using different stellar population model in our \\texttt{cluster\\_slug} analysis on the cluster physical properties. The use of non rotating and single star models produces a lower limit to the total estimated $Q(H^0)$ and therefore to $f_{esc}$, in particular at ages larger than 3-4 Myr. Binaries and rotating stars have the same effect, that is they increase the production of ionising photons at ages older than 3 Myr with respect to predictions from non-rotating single star models \\citep[see][]{leitherer14, gotberg19}.\nAny existing trend between $f_{esc}$ and age range of the stellar population would therefore be reinforced by different model assumptions, as also reported in the simulations of \\citet{ma20}.\n\n\\section{Conclusions}\n\\label{section:conclusions}\nWe have studied the ionised gas in the nearby galaxy NGC 7793 with MUSE, and complemented our observations with HST and ALMA data tracing the stellar content and the molecular gas.\n\n\\par We have constructed a census of YSCs, O stars, and WR stars in the MUSE FoV, and studied the properties of the stellar population. We modelled the age, mass, and ionising flux of YSCs with the stochastic stellar population synthesis code \\textsc{slug}, finding that clusters located in \\ion{H}{ii} regions have a higher probability to be younger, less massive, and to emit a higher number of ionising photons than clusters in the field.\n\n\\par We have investigated the link between the stellar population and the dense and ionised gas. We have contrasted the reddening map derived from individual stellar extinctions with the one constructed from the H$\\alpha$\/H$\\beta$ ratio and found that they compare well, but that the latter correlates better with the position of GMCs traced by ALMA. We have estimated an oxygen abundance for the \\ion{H}{ii} regions from the MUSE data, using the $S$-strong line method from \\citet{pilyugin16}. We found a median abundance of $12 + \\log(O\/H) \\sim 8.37$ with a scatter of 0.25 dex, in agreement with the previous estimate by \\citet{pilyugin14}. The abundance map appears to be rich in substructures, especially surrounding YSCs and WR stars. We caution however against possible degeneracies with for example the temperature and density, and we do indeed observe a correlation with the ionisation state of the regions.\n\n\\par We have studied the ionisation structure of the \\ion{H}{ii} regions using the [\\ion{S}{ii}]6716,31\/[\\ion{O}{iii}]4959,5007 ratio as a proxy for the optical depth. We have focused on a subset of regions and classified them based on the value of [\\ion{S}{ii}]\/[\\ion{O}{iii}] along their border into ionisation bounded or as featuring channels of optically thin gas. Finally, we have compiled a photoionisation budget for the entire FoV and for the subset of \\ion{H}{ii} regions.\n\n\\par Overall, we find an escape fraction $f_{esc} = 0.67_{-0.12}^{+0.08}$ for the population of \\ion{H}{ii} regions, and that the DIG in our FoV is more than consistent with being self-ionised, with an $f_{esc} = 0.87_{-0.05}^{+0.03}$. \nThis holds even when considering a lower-limit estimate for the DIG flux, derived by assuming a maximum mass of $30~M_\\odot$ for the field O stars. We furthermore find that the $f_{DIG, exp} \\gtrsim 0.17$ obtained by modelling the DIG stellar population is in good agreement with the DIG fraction derived from the observed H$\\alpha$ luminosity in Paper~\\textsc{i}, $f_{DIG, obs} = 0.15$. \n\n\\par We observe an $f_{esc} \\gtrsim 0.3$ in seven out of the eight studied regions, and investigate how the $f_{esc}$ is linked to the regions properties. Among others, we do not find any trend between $f_{esc}$ and the visual appearance of the regions; we read this as the effect of 3D geometry in our number limited sample.\n\n\\begin{acknowledgements}\nWe would like to thank the anonymous referee for the constructive feedback on the second draft of the manuscript.\nThis work is based on observations collected at the European Southern Observatory under ESO programme 60.A-9188(A).\nA.A. acknowledges the support of the Swedish Research Council, Vetenskapsr\\aa{}det, and the Swedish National Space Agency (SNSA).\nThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 757535).\nThis research made use of Astropy\\footnote{http:\/\/www.astropy.org}, a community-developed core Python package for Astronomy \\citep{astropy:2013, astropy:2018}.\n\\end{acknowledgements}\n\\bibpunct{(}{)}{;}{a}{}{,}\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n Currently,\n direct Dark Matter detection experiments\n searching for Weakly Interacting Massive Particles (WIMPs)\n are one of the promising methods\n for understanding the nature of Dark Matter (DM)\n and identifying them among new particles produced at colliders\n as well as\n studying the (sub)structure of our Galactic halo\n \\cite{SUSYDM96,\n       Drees12, Saab12, Baudis12c}.\n\n In our earlier work\n \\cite{DMDDf1v},\n we developed methods\n for reconstructing the (moments of the)\n time--averaged one--dimensional velocity distribution of halo WIMPs\n by using\n the measured recoil energies directly.\n This analysis requires\n no prior knowledge about\n the WIMP density near the Earth\n nor about their scattering cross section on nucleus,\n the unique required information\n is the mass of incident WIMPs.\n We therefore turned to develop the method for\n determining the WIMP mass model--independently\n by combining two experimental data sets\n with two different target nuclei\n \\cite{DMDDmchi}.\n By combining these methods\n and using two or three experimental data sets\n with different detector materials,\n one could reconstruct the one--dimensional velocity distribution\n of Galactic WIMPs directly.\n However,\n as presented in Ref.~\\cite{DMDDf1v},\n with a few hundreds or even thousands recorded WIMP events,\n only estimates of the reconstructed velocity distribution\n with pretty large statistical uncertainties\n at a few ($<$ 10) points\n could be obtained.\n\n Therefore,\n in order to offer more detailed information\n about the Galactic WIMP velocity distribution,\n we introduce in this paper the Bayesian analysis\n into our model--independent reconstruction procedure\n developed in Ref.~\\cite{DMDDf1v}\n to be able to determine,\n e.g.~the position of the peak of\n the one--dimensional velocity distribution function\n and the concrete values of\n the characteristic Solar and Earth's Galactic velocities.\n\n The remainder of this paper is organized as follows.\n In Sec.~2,\n we first review the model--independent method\n for reconstructing the time--averaged\n one--dimensional velocity distribution of halo WIMPs\n by using data from direct DM detection experiments directly.\n Then\n we introduce the Bayesian analysis\n and give the basic formulae\n needed in the extended reconstruction process.\n In Sec.~3,\n we present numerical results of\n the reconstructed WIMP velocity distribution functions\n based on Monte--Carlo simulations\n for different generating\n and fitting velocity distributions.\n Different input WIMP masses\n as well as\n impure (pseudo--)data sets\n mixed with (artificially added) unrejected background events\n will also be considered.\n We conclude in Sec.~4.\n Some technical details for our analysis\n will be given in Appendix.\n\\section{Formalism}\n In this section,\n we develop the formulae\n needed for our Bayesian reconstruction of\n the one--dimensional velocity distribution function of halo WIMPs $f_1(v)$\n by using direct Dark Matter detection data directly.\n\n We first review the model--independent method\n for reconstructing the time--averaged\n WIMP velocity distribution\n by using experimental data,\n i.e.~measured recoil energies,\n directly from direct detection experiments.\n These ``reconstructed data''\n (with estimated statistical uncertainties)\n will be used as input information\n for the further Bayesian analysis.\n Then,\n in the second part of this section,\n we review the basic concept of the Bayesian analysis\n and give the formulae\n needed in our extended reconstruction procedure.\n\n\\subsection{Model--independent reconstruction of\n            one--dimensional WIMP velocity distribution}\n In this subsection,\n we review briefly the method for reconstructing\n the one--dimensional WIMP velocity distribution\n from experimental data directly.\n Detailed derivations and discussions\n can be found in Ref.~\\cite{DMDDf1v}.\n\n\\subsubsection{From the recoil spectrum}\n The basic expression for the differential event rate\n for elastic WIMP--nucleus scattering is given by \\cite{SUSYDM96}:\n\\beq\n   \\dRdQ\n = \\calA \\FQ \\int_{\\vmin}^{\\vmax} \\bfrac{f_1(v)}{v} dv\n\\~.\n\\label{eqn:dRdQ}\n\\eeq\n Here $R$ is the direct detection event rate,\n i.e.~the number of events\n per unit time and unit mass of detector material,\n $Q$ is the energy deposited in the detector,\n $F(Q)$ is the elastic nuclear form factor,\n $f_1(v)$ is the one--dimensional velocity distribution function\n of the WIMPs impinging on the detector,\n $v$ is the absolute value of the WIMP velocity\n in the laboratory frame.\n The constant coefficient $\\calA$ is defined as\n\\beq\n        \\calA\n \\equiv \\frac{\\rho_0 \\sigma_0}{2 \\mchi \\mrN^2}\n\\~,\n\\label{eqn:calA}\n\\eeq\n where $\\rho_0$ is the WIMP density near the Earth\n and $\\sigma_0$ is the total cross section\n ignoring the form factor suppression.\n The reduced mass $\\mrN$ is defined by\n\\beq\n        \\mrN\n \\equiv \\frac{\\mchi \\mN}{\\mchi + \\mN}\n\\~,\n\\label{eqn:mrN}\n\\eeq\n where $\\mchi$ is the WIMP mass and\n $\\mN$ that of the target nucleus.\n Finally,\n $\\vmin$ is the minimal incoming velocity of incident WIMPs\n that can deposit the energy $Q$ in the detector:\n\\beq\n   \\vmin\n = \\alpha \\sqrt{Q}\n\\label{eqn:vmin}\n\\eeq\n with the transformation constant\n\\beq\n        \\alpha\n \\equiv \\sfrac{\\mN}{2 \\mrN^2}\n\\~,\n\\label{eqn:alpha}\n\\eeq\n and $\\vmax$ is the maximal WIMP velocity\n in the Earth's reference frame,\n which is related to\n the escape velocity from our Galaxy\n at the position of the Solar system,\n $\\vesc~\\gsim~600$ km\/s.\n\n In our earlier work \\cite{DMDDf1v},\n it was found that,\n by using a time--averaged recoil spectrum $dR \/ dQ$\n and assuming that no directional information exists,\n the normalized one--dimensional velocity distribution function\n of incident WIMPs, $f_1(v)$, can be solved\n from Eq.~(\\ref{eqn:dRdQ}) directly as\n\\beq\n   f_1(v)\n = \\calN\n   \\cbrac{ -2 Q \\cdot \\dd{Q} \\bbrac{ \\frac{1}{\\FQ} \\aDd{R}{Q} } }\\Qva\n\\~,\n\\label{eqn:f1v_dRdQ}\n\\eeq \n where the normalization constant $\\calN$ is given by\n\\beq\n   \\calN\n = \\frac{2}{\\alpha}\n   \\cbrac{\\intz \\frac{1}{\\sqrt{Q}}\n                \\bbrac{ \\frac{1}{\\FQ} \\aDd{R}{Q} } dQ}^{-1}\n\\~.\n\\label{eqn:calN_int}\n\\eeq\n Here the integral\n goes over the entire physically allowed range of recoil energies:\n starting at $Q = 0$,\n and the upper limit of the integral has been written as $\\infty$.\n Note that,\n because $f_1(v)$ in Eq.~(\\ref{eqn:f1v_dRdQ})\n is the normalized velocity distribution,\n the normalization constant $\\cal N$ here is independent of\n the constant coefficient $\\cal A$\n defined in Eq.~(\\ref{eqn:calA}).\n Hence,\n as the most important consequence,\n the velocity distribution function of halo WIMPs\n reconstructed by Eq.~(\\ref{eqn:f1v_dRdQ})\n is independent of the local WIMP density $\\rho_0$\n as well as\n of the WIMP--nucleus cross section $\\sigma_0$.\n However,\n not only the overall normalization constant $\\calN$\n given in Eq.~(\\ref{eqn:calN_int}),\n but also the shape of the velocity distribution,\n through the transformation $Q = v^2 \/ \\alpha^2$\n in Eq.~(\\ref{eqn:f1v_dRdQ}),\n depends on the WIMP mass $\\mchi$\n (involved in the coefficient $\\alpha$\n  defined in Eq.~(\\ref{eqn:alpha})).\n\n\\subsubsection{From experimental data directly}\n In order to use the expressions\n (\\ref{eqn:f1v_dRdQ}) and (\\ref{eqn:calN_int})\n for reconstructing $f_1(v)$,\n one needs a functional form for the recoil spectrum $dR \/ dQ$.\n In practice\n this requires usually a fit to experimental data.\n However,\n data fitting will re--introduce some model dependence\n and make the error analysis more complicated.\n Hence,\n expressions that allow to reconstruct $f_1(v)$\n directly from data\n (i.e.~measured recoil energies)\n have also been developed \\cite{DMDDf1v}.\n We started by considering experimental data described by\n\\beq\n     {\\T Q_n - \\frac{b_n}{2}}\n \\le \\Qni\n \\le {\\T Q_n + \\frac{b_n}{2}}\n\\~,\n     ~~~~~~~~~~~~\n     i\n =   1,~2,~\\cdots,~N_n,~\n     n\n =   1,~2,~\\cdots,~B.\n\\label{eqn:Qni}\n\\eeq\n Here the entire experimental possible\n energy range between the minimal and maximal cut--offs\n $\\Qmin$ and $\\Qmax$\n has been divided into $B$ bins\n with central points $Q_n$ and widths $b_n$.\n In each bin,\n $N_n$ events will be recorded.\n\n As argued in Ref.~\\cite{DMDDf1v},\n the statistical uncertainty on\n the ``slope of the recoil spectrum'',\n $\\bbrac{d\/dQ \\~ (dR \/ dQ)}_{Q = Q_n}$,\n appearing in the expression (\\ref{eqn:f1v_dRdQ}),\n scales like the bin width to the power $-1.5$.\n In addition,\n the wider the bin width,\n the more the recorded events in this bin,\n and thus the smaller the statistical uncertainty\n on the estimator of $\\bbrac{d\/dQ \\~ (dR \/ dQ)}_{Q = Q_n}$.\n Hence,\n since the recoil spectrum $dR \/ dQ$ is expected\n to be approximately exponential \\cite{DMDDf1v},\n in order to approximate the spectrum\n in a rather wider range,\n instead of the conventional standard linear approximation,\n the following exponential ansatz\n for the measured recoil spectrum\n (before normalized by the exposure $\\calE$)\n in the $n$th bin has been introduced \\cite{DMDDf1v}:\n\\beq\n        \\adRdQ_{{\\rm expt}, \\~ n}\n \\equiv \\adRdQ_{{\\rm expt}, \\~ Q \\simeq Q_n}\n \\equiv \\rn  \\~ e^{k_n (Q - Q_{s, n})}\n\\~.\n\\label{eqn:dRdQn}\n\\eeq\n Here $r_n$ is the standard estimator\n for $(dR \/ dQ)_{\\rm expt}$ at $Q = Q_n$:\n\\beq\n   r_n\n = \\frac{N_n}{b_n}\n\\~,\n\\label{eqn:rn}\n\\eeq\n $k_n$ is the logarithmic slope of\n the recoil spectrum in the $n$th $Q-$bin,\n which can be computed numerically\n from the average value of the measured recoil energies\n in this bin:\n\\beq\n   \\bQn\n = \\afrac{b_n}{2} \\coth\\afrac{k_n b_n}{2}-\\frac{1}{k_n}\n\\~,\n\\label{eqn:bQn}\n\\eeq\n where\n\\beq\n        \\bQxn{\\lambda}\n \\equiv \\frac{1}{N_n} \\sumiNn \\abrac{\\Qni - Q_n}^{\\lambda}\n\\~.\n\\label{eqn:bQn_lambda}\n\\eeq\n Then the shifted point $Q_{s, n}$\n in the ansatz (\\ref{eqn:dRdQn}),\n at which the leading systematic error\n due to the ansatz\n is minimal \\cite{DMDDf1v},\n can be estimated by\n\\beq\n   Q_{s, n}\n = Q_n + \\frac{1}{k_n} \\ln\\bfrac{\\sinh(k_n b_n\/2)}{k_n b_n\/2}\n\\~.\n\\label{eqn:Qsn}\n\\eeq\n Note that $Q_{s, n}$ differs from the central point of the $n$th bin, $Q_n$.\n\n Now,\n substituting the ansatz (\\ref{eqn:dRdQn})\n into Eq.~(\\ref{eqn:f1v_dRdQ})\n and then letting $Q = Q_{s, n}$,\n we can obtain that\n\\beq\n   f_{1, {\\rm rec}}(v_{s, n})\n = \\calN\n   \\bBigg{\\frac{2 Q_{s, n} r_n}{F^2(Q_{s, n})}}\n   \\bbrac{\\dd{Q} \\ln \\FQ \\bigg|_{Q = Q_{s, n}} - k_n}\n\\~.\n\\label{eqn:f1v_Qsn}\n\\eeq\n Here\n\\beq\n   v_{s, n}\n = \\alpha \\sqrt{Q_{s, n}}\n\\~,\n\\label{eqn:vsn}\n\\eeq\n and the normalization constant $\\calN$\n given in Eq.~(\\ref{eqn:calN_int})\n can be estimated directly from the data by\n\\beq\n   \\calN\n = \\frac{2}{\\alpha}\n   \\bbrac{\\sum_{a} \\frac{1}{\\sqrt{Q_a} \\~ F^2(Q_a)}}^{-1}\n\\~,\n\\label{eqn:calN_sum}\n\\eeq\n where the sum runs over all events in the sample.\n\n\\subsubsection{Windowing the data set}\n As mentioned above,\n the statistical uncertainty on\n the slope of the recoil spectrum\n around the central point $Q_n$,\n $\\bbrac{d\/dQ \\~ (dR \/ dQ)}_{Q \\simeq Q_n}$,\n is approximately proportional to $b_n^{-1.5}$.\n Thus,\n in order to reduce the statistical uncertainty\n on the velocity distribution\n reconstructed by Eq.~(\\ref{eqn:f1v_Qsn}),\n it seems to be better\n to use large bin width.\n However,\n neither the conventional linear approximation:\n\\beq\n   \\adRdQ_{{\\rm expt}, \\~ Q = Q_n}\n = \\frac{N_n}{b_n}\n\\eeq\n nor the exponential ansatz\n given in Eq.~(\\ref{eqn:dRdQn})\n can describe the real (but as yet unknown)\n recoil spectrum exactly.\n The neglected terms of higher powers of $Q - Q_n$\n could therefore induce some uncontrolled systematic errors\n which increase with increasing bin width.\n Moreover,\n since the number of bins scales inversely with their size,\n by using larger bins we would be able to estimate $f_1(v)$\n only at a smaller number of velocities.\n Additionally,\n once a quite large bin width is used,\n it would correspondingly lead to a quite large value\n of the first reconstructible point of $f_1(v)$,\n i.e.~$f_{1, {\\rm rec}}(v_{s, 1})$,\n since the central point $Q_1$\n as well as\n the shifted point $Q_{s, 1}$\n of the first bin would be quite large.\n Finally,\n choosing a fixed bin size,\n as one conventionally does,\n would let errors on the estimated logarithmic slopes,\n and hence also on the estimates of $f_1(v)$,\n increase quickly with increasing $Q$ or $v$.\n This is due to the essentially exponential form\n of the expected recoil spectrum,\n which would lead to a quickly falling number of events\n in equal--sized bins.\n By some trial--and--error analyses\n it was found that\n the errors are roughly equal in all bins\n if the bin widths increase linearly \\cite{DMDDf1v}.\n\n Therefore,\n it has been introduced in Ref.~\\cite{DMDDf1v} that\n one can first collect experimental data\n in relatively small bins\n and then combining varying numbers of bins\n into overlapping ``windows''.\n In particular,\n the first window would be identical with the first bin.\n One starts by binning the data,\n as in Eq.~(\\ref{eqn:Qni}),\n where the bin widths satisfy\n\\beq\n   b_n\n = b_1 + (n - 1) \\delta\n\\~,\n\\label{eqn:bn_delta}\n\\eeq\n i.e.\n\\beq\n   Q_n\n = \\Qmin + \\abrac{n - \\frac{1}{2}} b_1 + \\bfrac{(n - 1)^2}{2} \\delta\n\\~.\n\\label{eqn:Qn_delta}\n\\eeq\n Here the increment $\\delta$ satisfies\n\\beq\n   \\delta\n = \\frac{2}{B (B - 1)} \\aBig{\\Qmax - \\Qmin - B b_1}\n\\~,\n\\label{eqn:delta_B}\n\\eeq\n $B$ being the total number of bins,\n and $Q_{\\rm (min, max)}$ are\n the experimental minimal and maximal cut--off energies.\n Assume up to $n_W$ bins are collected into a window,\n with smaller windows at the borders of the range of $Q$.\n\n In order to distinguish the numbers of bins and windows,\n hereafter Latin indices $n,~m,~\\cdots$ are used to label bins,\n and Greek indices $\\mu,~\\nu,~\\cdots$ to label windows.\n For $1 \\leq \\mu \\leq n_W$,\n the $\\mu$th window simply consists of the first $\\mu$ bins;\n for $n_W \\leq \\mu \\leq B$,\n the $\\mu$th window consists of bins\n $\\mu-n_W + 1,~\\mu-n_W + 2,~\\cdots,~\\mu$;\n and for $B \\leq \\mu \\leq B+n_W-1$,\n the $\\mu$th window consists of the last $n_W - (\\mu - B)$ bins.\n This can also be described by introducing\n the indices $n_{\\mu-}$ and $n_{\\mu+}$\n which label the first and last bins\n contributing to the $\\mu$th window,\n with\n\\cheqna\n\\beq\n\\renewcommand{\\arraystretch}{1.3}\n   n_{\\mu-}\n = \\cleft{\\begin{array}{l c l}\n           1,         & ~~~~~~ & {\\rm for}~\\mu \\leq n_W, \\\\\n           \\mu-n_W+1, &        & {\\rm for}~\\mu \\geq n_W,\n          \\end{array}}\n\\label{eqn:n_mu_minus}\n\\eeq\n and\n\\cheqnb\n\\beq\n\\renewcommand{\\arraystretch}{1.3}\n   n_{\\mu+}\n = \\cleft{\\begin{array}{l c l}\n           \\mu, & ~~~~~~ & {\\rm for}~\\mu \\leq B, \\\\\n           B,   &        & {\\rm for}~\\mu \\geq B.\n          \\end{array}}\n\\label{eqn:n_mu_plus}\n\\eeq\n\\cheqn\n The total number of windows\n defined through Eqs.~(\\ref{eqn:n_mu_minus}) and (\\ref{eqn:n_mu_plus})\n is evidently $W = B + n_W - 1$,\n i.e.~$1 \\leq \\mu \\leq B + n_W - 1$.\n\n As shown above,\n the basic observables needed\n for the reconstruction of $f_1(v)$ by Eq.~(\\ref{eqn:f1v_Qsn})\n are the number of events in the $n$th $Q-$bin, $N_n$,\n as well as\n the average value of the measured recoil energies\n in this bin, $\\bQn$.\n For a ``windowed'' data set,\n one can easily calculate\n the number of events per window as\n\\beq\n   N_{\\mu}\n = \\sum_{n = n_{\\mu-}}^{n_{\\mu+}} N_n\n\\~,\n\\label{eqn:N_mu}\n\\eeq\n as well as\n the average value of the measured recoil energies\n\\beq\n   \\Bar{Q - Q_{\\mu}}|_{\\mu}\n = \\frac{1}{N_{\\mu}}\n   \\abrac{\\sum_{n = n_{\\mu-}}^{n_{\\mu+}} N_n \\Bar{Q}|_{n}} - Q_{\\mu}\n\\~,\n\\label{eqn:wQ_mu}\n\\eeq\n where $Q_{\\mu}$ is the central point of the $\\mu$th window.\n The exponential ansatz in Eq.~(\\ref{eqn:dRdQn})\n is now assumed to hold over an entire window.\n We can then estimate the prefactor as\n\\beq\n   r_{\\mu}\n = \\frac{N_{\\mu}}{w_{\\mu}}\n\\~,\n\\label{eqn:r_mu}\n\\eeq\n $w_{\\mu}$ being the width of the $\\mu$th window.\n The logarithmic slope of the recoil spectrum\n in the $\\mu$th window, $k_{\\mu}$,\n as well as\n the shifted point $Q_{s, \\mu}$\n (from the central point of each ``window'', $Q_{\\mu}$)\n can be calculated as in Eqs.~(\\ref{eqn:bQn}) and (\\ref{eqn:Qsn})\n with ``bin'' quantities replaced by ``window'' quantities.\n Finally,\n note that,\n due to the combination of bins\n into overlapping windows,\n these quantities are all correlated (for $n_W \\neq 1$).\n The covariance matrix of\n the estimates of $f_1(v)$\n at adjacent values of $v_{s, \\mu} = \\alpha \\sqrt{Q_{s, \\mu}}$\n is given b\n\\footnote{\n Note that\n Eq.~(\\ref{eqn:cov_f1v_Qs_mu}) should in principle\n also include contributions involving\n the statistical error on the estimator for $\\calN$\n in Eq.~(\\ref{eqn:calN_sum}).\n However,\n this error and its correlations\n with the errors on the $r_{\\mu}$ and $k_{\\mu}$\n have been found to be negligible\n compared to the errors included in Eq.~(\\ref{eqn:cov_f1v_Qs_mu})\n \\cite{DMDDf1v}.\n}\n\\beqn\n \\conti {\\rm cov}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}), f_{1, {\\rm rec}}(v_{s, \\nu})}\n        \\non\\\\\n \\=     \\bfrac{f_{1, {\\rm rec}}(v_{s, \\mu}) f_{1, {\\rm rec}}(v_{s, \\nu})}\n              {r_{\\mu} r_{\\nu}}\n        {\\rm cov}\\abrac{r_{\\mu}, r_{\\nu}}\n       +\\abrac{2 \\calN}^2\n        \\bfrac{Q_{s, \\mu} Q_{s, \\nu} r_{\\mu} r_{\\nu}}{F^2(Q_{s, \\mu}) F^2(Q_{s, \\nu})}\n        {\\rm cov}\\abrac{k_{\\mu}, k_{\\nu}}\n        \\non\\\\\n\\conti ~~~~~~~~~~~~\n       -\\calN\n        \\cbrac{ \\bfrac{f_{1, {\\rm rec}}(v_{s, \\mu})}{r_{\\mu}}\n                \\bfrac{2 Q_{s, \\nu} r_{\\nu} }{F^2(Q_{s, \\nu})}\n                {\\rm cov}\\abrac{r_{\\mu}, k_{\\nu}}\n               +\\aBig{\\mu \\lgetsto \\nu}}\n\\~.\n\\label{eqn:cov_f1v_Qs_mu}\n\\eeqn\n\n\\subsection{Bayesian analysis}\n In this subsection,\n we review the basic concept of the Bayesian analysis\n \\cite{Barlow-Statistics}\n (for its applications in physics,\n  see e.g.~Ref.~\\cite{DAgostini95},\n  and for its recent applications\n  in direct DM detection phenomenology,\n  see e.g.~Refs.~\\cite{Akrami,\n                       Pato,\n                       Arina,\n                       Strege12,\n                       Cerdeno13,\n                       Kavanagh13})\n and extend this method to the use of\n the ``reconstructed data''\n offered by our model--independent reconstruction process\n described in the previous subsection.\n\n\\subsubsection{Baye's theorem}\n We start with the {\\em Baye's theorem}:\n the probability of $A$ given that $B$ is true\n multiplies the probability of that $B$ is true\n is equal to\n the probability of that\n $A$ and $B$ happen simultaneously,\n which\n is also equal to\n the probability of $B$ given that $A$ is true\n multiplies the probability of that $A$ is true.\n This can simply be expressed as\n\\beq\n    {\\rm p}(A | B) \\~ {\\rm p}(B)\n =  {\\rm p}(A \\cap B)\n =  {\\rm p}(B | A) \\~ {\\rm p}(A)\n\\~,\n\\eeq\n where ${\\rm p}(A | B)$ is called\n the ``conditional probability'' of $A$\n given that $B$ is true.\n As long as ${\\rm p}(B) \\neq 0$,\n the above equation can be rewritten as\n\\beqn\n     {\\rm p}(A | B)\n \\=  \\frac{{\\rm p}(B | A) \\~ {\\rm p}(A)}{{\\rm p}(B)}\n     \\non\\\\\n \\=  \\frac{{\\rm p}(B | A) \\~ {\\rm p}(A)}\n          {{\\rm p}(B | A) \\~ {\\rm p}(A) + {\\rm p}(B | \\Bar{A}) \\~ {\\rm p}(\\Bar{A})}\n\\~,\n\\label{eqn:Bayes_theorem}\n\\eeqn\n where\n\\(\n     {\\rm p}(\\Bar{A})\n =   1 - {\\rm p}(A)\n\\)\n is the probability of the complement of $A$,\n i.e.~the probability of that\n $A$ is not happen.\n\n\\subsubsection{Bayesian statistics}\n Applying the Baye's theorem\n described\n above,\n one can directly have that\n\\beq\n     {\\rm p}({\\rm theory} | {\\rm result})\n  =  \\frac{{\\rm p}({\\rm result} | {\\rm theory})}\n          {{\\rm p}({\\rm result})} \\cdot {\\rm p}({\\rm theory})\n\\~.\n\\label{eqn:Bayesian_statistics}\n\\eeq\n This means that,\n given the observed result,\n the probability of that\n a specified theory is true\n is proportional to\n the probability of the observed result\n in the specified theory\n multiplies\n the probability of the specified theory.\n\n This statement can be understood as follows.\n If the observed result is predicted by a specified theory\n to be highly unlikely or even impossible\/forbidden,\n then this observation\n makes the ``degree of belief'' of this specified theory small\n or even disproves this theory.\n In contrast,\n the observation of a prediction\n by a specified theory with a high probability\n will strengthen one's belief in this theory.\n\n\\subsubsection{Bayesian analysis}\n By extending Eq.~(\\ref{eqn:Bayesian_statistics}),\n we can obtain that\n\\beq\n     {\\rm p}(\\Theta | {\\rm data})\n  =  \\frac{{\\rm p}({\\rm data} | \\Theta)}\n          {{\\rm p}({\\rm data})} \\cdot {\\rm p}(\\Theta)\n\\~.\n\\label{eqn:Bayesian_analysis}\n\\eeq\n Here\n $\\Theta = \\cbig{a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}$\n denotes a specified (combination of the) value(s)\n of the fitting parameter(s);\n ${\\rm p}(\\Theta)$,\n called ``prior probability'',\n represents our degree of belief about\n $\\Theta$ being the true value(s) of fitting parameter(s),\n which is often given\n in form of the (multiplication of the)\n probability distribution(s) of the fitting parameter(s).\n ${\\rm p}({\\rm data})$,\n called ``evidence'',\n is the total probability of obtaining the particular set of data,\n which is in practice\n irrespective of the actual value(s) of the\n parameter(s) and\n can be treated as a normalization constant;\n it will not be of interest in our further discussion.\n ${\\rm p}({\\rm data} | \\Theta)$\n denotes the probability of the observed result,\n once the specified (combination of the) value(s)\n of the fitting parameter(s)\n happens,\n which can usually be described\n by the {\\em likelihood} function of $\\Theta$,\n ${\\cal L}(\\Theta)$.\n Finally,\n ${\\rm p}(\\Theta | {\\rm data})$,\n called the ``posterior probability density function''\n for $\\Theta$,\n representes\n the probability of that\n the specified (combination of the) value(s)\n of the fitting parameter(s)\n happens,\n given the observed result.\n\n\\subsubsection{Bayesian reconstruction of \\boldmath$f_1(v)$}\n Now,\n we can describe\n the procedure of our Bayesian reconstruction of\n the one--dimensional WIMP velocity distribution function\n in detail.\n\n First,\n by using Eqs.~(\\ref{eqn:f1v_Qsn})\n to (\\ref{eqn:calN_sum}),\n one can obtain $W = B + n_W - 1$ reconstructed data points:\n $\\aBig{v_{s, \\mu}, f_{1, {\\rm rec}}(v_{s, \\mu}) \\pm \\sigma_{f_1, s, \\mu}}$,\n for $\\mu = 1,~2,~\\cdots,~W$,\n where\n\\beq\n        \\sigma_{f_1, s, \\mu}\n \\equiv \\sqrt{{\\rm cov}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}), f_{1, {\\rm rec}}(v_{s, \\mu})}}\n\\label{eqn:sigma_f1v_Qs_mu}\n\\eeq\n denote the square roots of\n the diagonal entries of the covariance matrix\n given in Eq.~(\\ref{eqn:cov_f1v_Qs_mu}).\n Choosing a theoretical prediction of\n the one--dimensional velocity distribution\n of halo WIMPs:\n $f_{1, \\rm th}(v; a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})$,\n where $\\abig{a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}$\n are the $N_{\\rm Bayesian}$ fitting parameters,\n and\n assuming that\n the reconstructed data points\n is {\\em Gaussian}--distributed\n around the theoretical predictions\n $f_{1, \\rm th}(v_{s, \\mu}; a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})$,\n the likelihood function for ${\\rm p}({\\rm data} | \\Theta)$\n can then be defined by\n\\beqn\n \\conti\n     {\\cal L}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W;~\n                   a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}\n     \\non\\\\\n \\eqnequiv\n     \\prod_{\\mu = 1}^{W}\n     {\\rm Gau}\\aBig{v_{s, \\mu},\n                    f_{1, {\\rm rec}}(v_{s, \\mu}),\n                    \\sigma_{f_1, s, \\mu};\n                    a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}\n\\~,\n\\label{eqn:calL}\n\\eeqn\n where\n\\beqn\n \\conti\n     {\\rm Gau}\\aBig{v_{s, \\mu},\n                    f_{1, {\\rm rec}}(v_{s, \\mu}),\n                    \\sigma_{f_1, s, \\mu};\n                    a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}\n     \\non\\\\\n \\eqnequiv\n     \\frac{1}{\\sqrt{2 \\pi} \\~ \\sigma_{f_1, s, \\mu}} \\~\n     e^{-\\bbig{  f_{1, {\\rm rec}}(v_{s, \\mu})\n               - f_{1, \\rm th}(v_{s, \\mu}; a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})}^2 \\big\/\n         2 \\sigma_{f_1, s, \\mu}^2}\n\\~.\n\\label{eqn:Bayesian_DF_Gau}\n\\eeqn\n Or,\n equivalently,\n we can use the {\\em logarithmic} likelihood function\n given by\n\\beqn\n \\conti\n    \\ln{\\cal L}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W;~\n                     a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}\n     \\non\\\\\n \\=- \\frac{1}{2}\n     \\sum_{\\mu = 1}^{W}\n     \\frac{\\bBig{  f_{1, {\\rm rec}}(v_{s, \\mu})\n                 - f_{1, \\rm th}(v_{s, \\mu}; a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})}^2}\n          {\\sigma_{f_1, s, \\mu}^2}\n   - \\sum_{\\mu = 1}^{W}\n     \\ln \\abrac{\\sqrt{2 \\pi} \\~ \\sigma_{f_1, s, \\mu}}\n\\~.\n\\label{eqn:ln_calL}\n\\eeqn\n Note that\n in practical use\n the second term in Eq.~(\\ref{eqn:ln_calL}) can be neglected,\n since it is just a constant\n for all scanned (combinations of) values of\n the fitting parameter(s)\n $\\abig{a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}}}$.\n\n Finally,\n choosing the probability distribution function\n for each fitting parameter $a_i$,\n ${\\rm p}_i(a_i)$,\n the posterior probability density\n on the left--hand side of Eq.~(\\ref{eqn:Bayesian_analysis})\n can then be given by\n\\beqn\n \\conti\n     {\\rm p}\\aBig{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}~\\Big|~\n                  f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W}\n     \\non\\\\\n \\eqnpropto\n     {\\cal L}\\aBig{f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W;~\n                   a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}\n     \\prod_{i = 1}^{N_{\\rm Bayesian}} {\\rm p}_i(a_i)\n\\~.\n\\label{eqn:P_Bayesian}\n\\eeqn\n\n\\section{Numerical results}\n In this section,\n we present numerical results\n of our Bayesian reconstruction of\n the one--dimensional velocity distribution function\n of halo WIMPs\n based on Monte--Carlo simulations.\n\n In order to test\n whether we can reconstruct $f_1(v)$ correctly\n with even an improper adopted model\n and\/or incorrect expectation value(s) of the fitting parameter(s),\n different choices of the WIMP velocity distribution function\n for generating signal events\n as well as\n different assumptions of $f_1(v)$ and\/or\n (slightly) different expectation value(s) of the fitting parameter(s)\n from the commonly adopted values\n for the Bayesian fitting process\n will be condisered in our simulations.\n In Tables \\ref{tab:setup_Gau},\n \\ref{tab:setup_sh} and \\ref{tab:setup_Gau_k},\n we list the {\\em input} setup\n for the chosen {\\em input} velocity distribution functions\n used for {\\em generating} WIMP signals\n as well as\n the scanning ranges,\n the expectation values of\n and 1$\\sigma$ uncertainties on\n the fitting parameters\n used for different {\\em fitting} velocity distributions.\n Additionally,\n a common maximal cut--off\n on the one--dimensional WIMP velocity distribution\n has been set as \\mbox{$\\vmax = 700$ km\/s}.\n\n The WIMP mass $\\mchi$\n involved in the coefficient $\\alpha$\n in Eqs.~(\\ref{eqn:vsn}) and (\\ref{eqn:calN_sum})\n for estimating the reconstructed points $v_{s, \\mu}$\n as well as\n the normalization constant $\\calN$\n has been assumed\n either to be known precisely with a negligible uncertainty\n from other (e.g.~collider) experiments\n or determined from\n direct detection experiments\n with {\\em different} data sets.\n As in Ref.~\\cite{DMDDf1v},\n a $\\rmXA{Ge}{76}$ nucleus has been chosen\n as our detector material for reconstructing $f_1(v)$,\n whereas a $\\rmXA{Si}{28}$ target\n and a {\\em second} $\\rmXA{Ge}{76}$ target\n have been used for determining $\\mchi$\n \\cite{DMDDmchi}.\n\n As in Refs.~\\cite{DMDDf1v, DMDDbg-f1v},\n the WIMP--nucleus cross section in Eq.~(\\ref{eqn:calA})\n has been assumed\n to be only spin--independent (SI),\n \\mbox{$\\sigmapSI = 10^{-9}$ pb},\n and\n the commonly used analytic form\n for the elastic nuclear form factor:\n\\beq\n   F_{\\rm SI}^2(Q)\n = \\bfrac{3 j_1(q R_1)}{q R_1}^2 e^{-(q s)^2}\n\\label{eqn:FQ_WS}\n\\eeq\n has been adopted.\n Here $Q$ is the recoil energy\n transferred from the incident WIMP to the target nucleus,\n $j_1(x)$ is a spherical Bessel function,\n\\(\n   q\n = \\sqrt{2 m_{\\rm N} Q}\n\\)\n is the transferred 3-momentum,\n for the effective nuclear radius we use\n\\(\n   R_1\n = \\sqrt{R_A^2 - 5 s^2}\n\\)\n with\n\\(\n        R_A\n \\simeq 1.2 \\~ A^{1\/3}~{\\rm fm}\n\\)\n and a nuclear skin thickness\n\\(\n        s\n \\simeq 1~{\\rm fm}\n\\).\n\n In our simulations,\n the experimental threshold energies\n have been assumed to be negligible ($\\Qmin = 0$\n\\footnote{\n Note that,\n once the experimental threshold energy\n of the data set\n used for reconstructing $f_1(v)$\n is non--negligible,\n the estimate (\\ref{eqn:calN_sum})\n of the normalization constant $\\calN$\n would need to be modified properly,\n especially for light WIMPs.\n}\n and the maximal cut--off energies\n are set as \\mbox{$\\Qmax = 100$ keV}\n for all target nucle\n\\footnote{\n Note that,\n due to the maximal cut--off\n on the one--dimensional WIMP velocity distribution,\n $\\vmax$,\n a kinematic maximal cut--off energy\n\\beq\n   Q_{\\rm max, kin}\n = \\frac{\\vmax^2}{\\alpha^2}\n\\label{eqn:Qmax_kin}\n\\eeq\n has also been taken into account.\n};\n the widths of the first energy bin in Eq.~(\\ref{eqn:bn_delta})\n have also been set commonly as \\mbox{$b_1 = 10$ keV}.\n Additionally,\n we assumed that\n all experimental systematic uncertainties\n as well as\n the uncertainty on the measurement of the recoil energy\n could be ignored.\n The energy resolution of most existing and next--generation detectors\n should be good enough so that\n the very small measurement uncertainties can be neglected\n compared to the statistical uncertainties\n on our reconstructed results\n with only few events.\n Energy range between $\\Qmin$ and $\\Qmax$\n have been divided into five bins\n and up to three bins\n have been combined to a window.\n (\\mbox{3 $\\times$}) 5,000 experiments with 500 total events on average\n in one experiment have been simulated.\n In Secs.~3.1 to 3.3,\n the input WIMP mass has been fixed as \\mbox{$\\mchi = 100$ GeV}.\n In Sec.~3.4,\n we consider the cases with\n a light WIMP mass of \\mbox{$\\mchi = 25$ GeV}\n and a heavy one of \\mbox{$\\mchi = 250$ GeV}.\n In addition,\n in Sec.~3.5,\n we consider also briefly\n the effects of unrejected background events\n for different input WIMP masses\n \\cite{DMDDbg-mchi, DMDDbg-f1v}.\n\n Note that,\n for our numerical simulations\n presented in this section,\n the actual numbers of generated signal and background events\n in each simulated experiment\n are Poisson--distributed around their expectation values\n {\\em independently}.\n This means that,\n for example,\n for simulations shown in Figs.~\\ref{fig:dRdQ-bg-ex-const-000-100-20-100}\n we generate 400 (100) events {\\em on average} for WIMP--signals (backgrounds)\n and the total event number recorded in one experiment\n is then the sum of these two numbers.\n\n Regarding our degree of belief about\n each fitting parameter $a_i$,\n i.e.~${\\rm p}_i(a_i)$ in Eq.~(\\ref{eqn:P_Bayesian}),\n two probability distribution functions\n have been considered.\n The simplest one is the flat--distribution:\n\\beq\n    {\\rm p}_i(a_i)\n =  1\n\\~,\n    ~~~~~~~~~~~~~~~~\n    {\\rm for~} a_{i, \\rm min} \\le a_i \\le a_{i, \\rm max},\n\\label{eqn:Bayesian_DF_a_flat}\n\\eeq\n where $a_{i, \\rm (min, max)}$ denote\n the minimal and maximal bounds of the scanning interval of\n the fitting parameter $a_i$.\n On the other hand,\n for the case that\n we have already prior knowledge about\n one fitting parameter,\n a Gaussian--distribution:\n\\beq\n    {\\rm p}_i(a_i; \\mu_{a, i}, \\sigma_{a, i})\n =  \\frac{1}{\\sqrt{2 \\pi} \\~ \\sigma_{a, i}} \\~\n    e^{-(a_i - \\mu_{a, i})^2 \/ 2 \\sigma_{a, i}^2}\n\\label{eqn:Bayesian_DF_a_Gau}\n\\eeq\n with the expectation value $\\mu_{a, i}$ of and\n the 1$\\sigma$ uncertainty $\\sigma_{a, i}$ on\n the fitting parameter $a_i$\n is used.\n\n Note that,\n in one simulated experiment,\n we scan the parameter space\n $(a_1, a_2, \\cdots, a_{N_{\\rm Bayesian}})$\n in the volume\n $a_i \\in [a_{i, {\\rm min}}, a_{i, {\\rm max}}]$,\n $i = 1, 2, \\cdots, N_{\\rm Bayesian}$\n to find a particular point\n $(a_1^{\\ast}, a_2^{\\ast}, \\cdots, a_{N_{\\rm Bayesian}}^{\\ast})$,\n which maximizes the (numerator of the)\n posterior probability density: \\\\\n ${\\rm p}\\aBig{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}~\\Big|~\n               f_{1, {\\rm rec}}(v_{s, \\mu}),~\\mu = 1,~2,~\\cdots,~W}$.\n After that\n all simulations have been done,\n we determine the median value of\n the (1$\\sigma$ lower and upper bounds of the) velocity distribution\n reconstructed by Eq.~(\\ref{eqn:f1v_Qsn}) from all experiments,\n denoted as\n $f_{1, {\\rm median}}\\abrac{\\alpha_{\\rm (median)} \\sqrt{Q_{s, \\mu, {\\rm Bayesian}}}}$,\n for $\\mu = 1,~2,~\\cdots,~W$,\n and shown as solid black crosses in the (top--)left frame(s) of,\n e.g.~Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}.\n And\n we define further\n\\beqn\n \\conti    {\\rm p}_{\\rm median}\\aBig{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}\n           \\non\\\\\n \\eqnequiv {\\rm p}\\aBig{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}~\\Big|~\n                        f_{1, {\\rm median}}\\abrac{\\alpha_{\\rm (median)} \\sqrt{Q_{s, \\mu, {\\rm Bayesian}}}},\n                        \\mu = 1,~2,~\\cdots,~W}\n\\~,\n           \\non\\\\\n\\label{eqn:P_Bayesian_median}\n\\eeqn\n and check the points\n $(a_1^{\\ast}, a_2^{\\ast}, \\cdots, a_{N_{\\rm Bayesian}}^{\\ast})$\n obtained from all simulated experiments\n to find the special (``best--fit'') point\n $(a_{1, {\\rm max}}, a_{2, {\\rm max}}, \\cdots, a_{N_{\\rm Bayesian}, {\\rm max}})$,\n which maximizes ${\\rm p}_{\\rm median}(a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian})$.\n\n\\subsection{Simple Maxwellian velocity distribution}\n\n\\begin{table}[t!]\n\\InsertSetupTable\n{simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{Simple\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n}\n{The input setup for\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n used for generating WIMP events\n as well as\n the scanning range,\n the expectation value of\n and the 1$\\sigma$ uncertainty on\n the unique fitting parameter $v_0$.\n}\n{tab:setup_Gau}\n\n We consider first\n the simplest isothermal spherical Galactic halo model\n for generating WIMP events.\n The normalized one--dimensional\n simple Maxwellian velocity distribution function\n can be expressed as\n \\cite{SUSYDM96, DMDDf1v}:\n\\beq\n   f_{1, \\Gau}(v)\n = \\frac{4}{\\sqrt{\\pi}}\n   \\afrac{v^2}{v_0^3} e^{-v^2 \/ v_0^2}\n\\~,\n\\label{eqn:f1v_Gau}\n\\eeq\n where $v_0 \\approx 220$ km\/s\n is the Solar orbital velocity in the Galactic frame.\n\n In Table \\ref{tab:setup_Gau},\n we list the input setup for\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n used for generating WIMP events\n as well as\n the scanning range,\n the expectation value of\n and the 1$\\sigma$ uncertainty on\n the unique fitting parameter $v_0$.\n Note that,\n for generating WIMP signals,\n \\mbox{$v_0 = 220$ km\/s} has been used,\n whereas for Bayesian fitting of this unique parameter\n we used a slightly different value of \\mbox{$v_0 = 230$ km\/s}\n and set a 1$\\sigma$ uncertainty\n of \\mbox{$\\sigma(v_0) = 20$ km\/s}.\n\n\\subsubsection{Simple Maxwellian velocity distribution}\n\n As the simplest Galactic halo model,\n we consider first the use of\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n given in Eq.~(\\ref{eqn:f1v_Gau})\n with an unique fitting parameter $v_0$\n to fit the reconstructed--input data\n given by Eqs.~(\\ref{eqn:f1v_Qsn}) to (\\ref{eqn:calN_sum})\n and (\\ref{eqn:cov_f1v_Qs_mu}).\n\n In Fig.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}(a),\n we show the reconstructed\n simple Maxwellian velocity distribution function\n for an input WIMP mass of \\mbox{$\\mchi = 100$ GeV}\n with a $\\rmXA{Ge}{76}$ target.\n Here the flat distribution\n given by Eq.~(\\ref{eqn:Bayesian_DF_a_flat})\n for the fitting parameter $v_0$\n has been used.\n The black crosses are the velocity distribution\n reconstructed by Eqs.~(\\ref{eqn:vsn}) and (\\ref{eqn:f1v_Qsn}):\n the vertical error bars show\n the square roots of the diagonal entries of the covariance matrix\n given in Eq.~(\\ref{eqn:cov_f1v_Qs_mu})\n (i.e.~$\\sigma_{f_1, s, \\mu}$\n  given in Eq.~(\\ref{eqn:sigma_f1v_Qs_mu}))\n and\n the horizontal bars indicate\n the sizes of the windows used\n for estimating $f_{1, {\\rm rec}}(v_{s, \\mu})$,\n respectively.\n The solid red curve\n is the {\\em generating} simple Maxwellian velocity distribution\n with an input value of \\mbox{$v_0 = 220$ km\/s}.\n While\n the dashed green curve indicates\n the {\\em reconstructed} simple Maxwellian velocity distribution\n with the fitting parameter $v_0$\n given by the {\\em median} value of all simulated experiments,\n the dash--dotted blue curve indicates\n the {\\em reconstructed} simple Maxwellian velocity distribution\n with\n $v_0$\n which maximizes ${\\rm p}_{\\rm median}\\abrac{a_i,~i = 1,~2,~\\cdots,~N_{\\rm Bayesian}}$\n defined in Eq.~(\\ref{eqn:P_Bayesian_median}).\n\n\\plotGeGauGauflat\n\\plotGeGauGauGau\n\n Meanwhile,\n the light--green (light--blue) area shown here\n indicate the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n of the Bayesian reconstructed velocity distribution function,\n which has been determined as follows.\n After scanning\n the reconstructed fitting parameter $v_0$\n obtained from all simulated experiments\n and ordering\n according to their ${\\rm p}_{\\rm median}$ values\n defined in Eq.~(\\ref{eqn:P_Bayesian_median})\n {\\em descendingly},\n we can not only\n determine the point\n which maximizes ${\\rm p}_{\\rm median}$\n (labeled with the subscript ``max'' in our plot\n\\footnote{\n Note that\n the subscript ``input'' in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}\n and \\ref{fig:f1v-Ge-100-0500-Gau-Gau-Gau}\n indicate that\n the WIMP mass needed\n in Eqs.~(\\ref{eqn:vsn}) and (\\ref{eqn:f1v_Qsn})\n has been given as the input WIMP mass;\n whereas,\n the subscript ``algo'' in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-flat}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-Gau}\n indicate that\n the WIMP mass\n is reconstructed by the algorithmic procedure\n developed in Ref.~\\cite{DMDDmchi}.\n}),\n but also\n the smallest and largest values of\n the first 68.27\\% (95.45\\%) of all these reconstructed $v_0$'s.\n We then use the smallest (largest) value of\n the first 68.27\\% (95.45\\%) reconstructed $v_0$'s\n to give the $1\\~(2)\\~\\sigma$ lower (upper) boundaries\n of the Bayesian reconstructed velocity distribution function.\n This means that\n all of the velocity distributions with $v_0$'s\n which give the largest 68.27\\% (95.45\\%) ${\\rm p}_{\\rm median}$ values\n should be in the $1\\~(2)\\~\\sigma$ light--green (light--blue) areas.\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 218.8 & $218.8\\~^{+12.6}_{-11.2}\\~(^{+33.6}_{-22.4})$ & [207.6, 231.4] & [196.4, 252.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 221.6 & $221.6 \\pm  9.8\\~(^{+23.8}_{-18.2})$ & [211.8, 231.4] & [203.4, 245.4] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 220.2 & $218.8\\~^{+21.0}_{-16.8}\\~(^{+49.0}_{-33.6})$ & [202.0, 239.8] & [185.2, 267.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 220.2 & $221.6\\~^{+15.4}_{-12.6}\\~(^{+33.6}_{-26.6})$ & [209.0, 237.0] & [195.0, 255.2] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_Gau}\n\n On the other hand,\n Fig.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}(b) shows\n the distribution of the Bayesian reconstructed fitting parameter $v_0$\n in all simulated experiments.\n The red vertical line indicates the true (input) value of $v_0$,\n which has been labeled with the subscript ``gen''.\n The green vertical line indicates\n the median value of the simulated results,\n whereas\n the blue one\n indicates the value which maximizes\n ${\\rm p}_{\\rm median}$.\n In addition,\n the horizontal thick (thin) green bars show\n the $1\\~(2)\\~\\sigma$ ranges of the reconstructed result\n\\footnote{\n Note that\n the $1\\~(2)\\~\\sigma$ ranges given here mean that,\n according to the order of the reconstructed values of $v_0$ {\\em along}\n and centered at their {\\em median} value $v_{0, {\\rm median}}$,\n 68.27\\% (95.45\\%) of the reconstructed values in the simulated experiments\n are in this range.\n}.\n Note that\n the bins at \\mbox{$v_0 = 160$ km\/s} and \\mbox{$v_0 = 300$ km\/s}\n are ``overflow'' bins,\n which contain also the experiments\n with the best--fit $v_0$ value of\n either \\mbox{$v_0 < 160$ km\/s} or \\mbox{$v_0 > 300$ km\/s}.\n\n In Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat},\n it can be seen clearly that,\n {\\em without} a prior knowledge about\n the Solar Galactic velocity,\n one could in principle pin down the parameter $v_0$ very precisely\n with $1\\~(2)\\~\\sigma$ statistical uncertainties of only\n \\mbox{$^{+12.6}_{-11.2}\\~(^{+33.6}_{-22.4})$ km\/s}\n (see Table \\ref{tab:results_Gau_Gau}).\n Moreover,\n by using the Bayesian reconstruction of\n the one--dimensional velocity distribution function,\n the large (1$\\sigma$) statistical uncertainty\n given by Eq.~(\\ref{eqn:sigma_f1v_Qs_mu})\n can be reduced significantly:\n the band of the 2$\\sigma$ statistical uncertainty\n would be approximately equal to or even smaller than\n the (solid black) vertical 1$\\sigma$ uncertainty bars!\n\n Furthermore,\n in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-Gau}\n we consider the case\n with a rough prior knowledge about the Solar Galactic velocity $v_0$.\n It has been found that,\n firstly,\n by using a Gaussian probability distribution\n for $v_0$ with a 1$\\sigma$ uncertainty of \\mbox{20 km\/s},\n one could reduce the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the Bayesian reconstructed parameter $v_0$\n to \\mbox{$\\pm  9.8\\~(^{+23.8}_{-18.2})$ km\/s}\n (see Table \\ref{tab:results_Gau_Gau}).\n Secondly and more importantly,\n although an expectation value of \\mbox{$v_0 = 230$ km\/s},\n which differs (slightly) from the true (input) one,\n is used,\n the value of the fitting parameter $v_0$\n could still be pinned down precisely\n with a tiny systematic deviation\n ($<$ 2 km\/s).\n\n\\plotGeSiGeGauGauflat\n\\plotGeSiGeGauGauGau\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-flat}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-Gau},\n we consider the case that\n the WIMP mass $\\mchi$\n needed in Eqs.~(\\ref{eqn:vsn}) and (\\ref{eqn:calN_sum})\n is reconstructed by the algorithmic procedure\n developed in Ref.~\\cite{DMDDmchi}\n with a $\\rmXA{Si}{28}$ target\n and a second $\\rmXA{Ge}{76}$ target.\n Note that,\n while the vertical bars show\n the 1$\\sigma$ statistical uncertainties\n estimated by Eq.~(\\ref{eqn:sigma_f1v_Qs_mu})\n taking into account an extra contribution from\n the 1$\\sigma$ statistical uncertainty\n on the reconstructed WIMP mass,\n the horizontal bars shown here\n indicate the 1$\\sigma$ statistical uncertainties\n on the estimates of $v_{s, \\mu}$\n given in Eq.~(\\ref{eqn:vsn})\n due to the uncertainty on the reconstructed WIMP mass;\n the statistical and systematic uncertainties\n due to estimating of $Q_{s, \\mu}$\n have been neglected here.\n\n It can be seen that,\n due to the extra {\\em statistical fluctuation}\n on the reconstructed WIMP mass\n \\cite{DMDDmchi}\n and in turn\n the uncertainty on the estimated $v_{s, \\mu}$\n (the horizontal bars in the left frames),\n the statistical uncertainties on the Bayesian reconstructed $v_0$\n with both of the flat and the Gaussian probability distributions\n become \\mbox{$\\sim 30$\\%} to \\mbox{$\\sim 60$\\%} larger.\n However,\n same as the case with an input WIMP mass,\n the use of the Gaussian probability distribution\n can not only reduce\n the statistical uncertainty on $v_0$ significantly\n and therefore improve\n the Bayesian reconstructed velocity distribution,\n but also alleviate\n the ``imprecisely'' expected value of $v_0$.\n\n In Table \\ref{tab:results_Gau_Gau},\n we list the reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the {\\em median} values of $v_0$.\n It would be worth to emphasize that,\n the statistical uncertainties shown\n in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-Gau}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-Gau}\n are (much) smaller than the {\\em input} uncertainty\n on the expectation value of $v_0$ of \\mbox{20 km\/s}.\n\n\\subsection{Shifted Maxwellian velocity distribution}\n\n\\begin{table}[t!]\n\\InsertSetupTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{Simple\n & $v_0$      [km\/s] & $\\sim 295$ & [160,    400] &   280 & 40   \\\\\n \\hline\n 1--para. shifted\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n \\hline\n \\multirow{2}{*}{Shifted}\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n & $\\ve$      [km\/s] &        231 & [160,    300] &   245 & 20   \\\\\n \\hline\n \\multirow{2}{*}{Variated shifted}\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n & $\\Delta v$ [km\/s] &         11 & [$-50$,   80] &    15 & 20   \\\\\n}\n{The input setup for\n the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n used for generating WIMP signals\n as well as\n the theoretically estimated values,\n the scanning ranges,\n the expectation values of\n and the 1$\\sigma$ uncertainties on\n the fitting parameters\n used for different fitting velocity distribution functions.\n}\n{tab:setup_sh}\n\n By taking into account\n the orbital motion of the Solar system around our Galaxy\n as well as\n that of the Earth around the Sun,\n a more realistic\n shifted Maxwellian velocity distribution of halo WIMPs\n has been given by\n \\cite{SUSYDM96, DMDDf1v}:\n\\beq\n   f_{1, \\sh}(v)\n = \\frac{1}{\\sqrt{\\pi}} \\afrac{v}{v_0 \\ve}\n   \\bBig{  e^{-(v - \\ve)^2 \/ v_0^2}\n         - e^{-(v + \\ve)^2 \/ v_0^2}  }\n\\~.\n\\label{eqn:f1v_sh}\n\\eeq\n Here\n $\\ve$ is the time-\u2013dependent\n Earth's velocity in the Galactic frame\n \\cite{Freese,\n       SUSYDM96}:\n\\beq\n   \\ve(t)\n = v_0 \\bbrac{1.05 + 0.07 \\cos\\afrac{2 \\pi (t - t_{\\rm p})}{1~{\\rm yr}}}\n\\~,\n\\label{eqn:ve}\n\\eeq\n with $t_{\\rm p} \\simeq$ June 2nd is the date\n on which the velocity of the Earth relative to the WIMP halo is maxima\n\\footnote{\n In our simulations,\n the time dependence of the Earth's velocity in the Galactic frame,\n the second term of $\\ve(t)$,\n will be ignored,\n i.e.~$\\ve = 1.05 \\~ v_0$\n is used.\n}.\n\n In Table \\ref{tab:setup_sh},\n we list the input setup for\n the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n used for generating WIMP signals\n as well as\n the theoretically estimated values,\n the scanning ranges,\n the expectation values of\n and the 1$\\sigma$ uncertainties on\n the fitting parameters \n used for different fitting velocity distribution functions.\n\n\\subsubsection{Simple Maxwellian velocity distribution}\n\n\\plotGeSiGeshGauflat\n\n\\plotGeSiGeshGauGau\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 296.8 & $296.8\\~^{+21.6}_{-19.2}\\~(^{+45.6}_{-36.0})$ & [277.6, 318.4] & [260.8, 342.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 294.4 & $292.0\\~^{+16.8}_{-12.0}\\~(^{+31.8}_{-26.4})$ & [280.0, 308.8] & [265.6, 323.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 299.2 & $296.8\\~^{+40.8}_{-31.2}\\~(^{+86.4}_{-60.0})$ & [265.6, 337.6] & [236.8, 383.2] \\\\\n \\cline{3-7}\n & & Gaussian &\n 294.4 & $294.4 \\pm 26.4\\~(\\pm 52.8)$ & [268.0, 320.8] & [241.6, 347.2] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_sh_Gau}\n\n We consider first the simplest case of\n the simple Maxwellian velocity distribution function $f_{1, \\Gau}(v)$\n with the unique fitting parameter $v_0$\n to fit the reconstructed--input data points\n given by Eqs.~(\\ref{eqn:f1v_Qsn}) and (\\ref{eqn:cov_f1v_Qs_mu}).\n\n As in Sec.~3.1.1,\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}\n we use first the flat probability distribution\n for the fitting parameter $v_0$\n with either the precisely known (input) (upper)\n or the reconstructed (lower) WIMP mass,\n respectively.\n Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(a)\n and\n (c)\n show clearly that,\n although an ``improper'' choice for\n the fitting velocity distribution function\n and a simple flat probability distribution for $v_0$\n (i.e.~without any prior knowledge about $v_0$)\n have been used,\n the 1$\\sigma$ statistical uncertainty bands\n of the reconstructed WIMP velocity distribution function\n could in principle still cover the true (input) distribution.\n More precisely and quantitatively,\n the deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n are only \\mbox{$\\sim 10$ km\/s}.\n Note that\n the 1$\\sigma$ statistical uncertainty\n on the reconstructed fitting parameter $v_0$\n is \\mbox{$\\sim 20$ km\/s} or even \\mbox{$\\sim 40$ km\/s}\n (see Table \\ref{tab:results_sh_Gau}).\n\n However,\n our simulations show also that,\n with an ``improper'' assumption about\n the fitting velocity distribution function,\n one would obtain an ``unexpected'' result\n for the fitting parameter $v_0$:\n 2.5$\\sigma$\n (with the reconstructed WIMP mass,\n  Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(d))\n to 4$\\sigma$\n (with the input WIMP mass,\n  Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(b))\n deviations of the\n reconstructed Solar Galactic velocity\n from the theoretical estimate of\n \\mbox{$v_0 \\approx 220$ km\/s}.\n Such observation would indicate clearly that\n our initial assumption about\n the fitting velocity distribution function\n would be incorrect \n or at least need to be modified.\n\n Moreover,\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-Gau}\n we assume that\n a rough prior knowledge about\n the Solar Galactic velocity $v_0$ exists\n and use the Gaussian probability distribution\n for $v_0$\n with an expectation value of \\mbox{$v_0 = {\\it 280}$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}.\n As observed in Sec.~3.1.1,\n with a prior knowledge about the fitting parameter $v_0$,\n one could reconstruct the velocity distribution function better:\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$\n could be reduced to \\mbox{$\\sim 60$\\%}.\n Additionally,\n the reconstructed 1$\\sigma$ statistical uncertainties on $v_0$\n with both of the input and the reconstructed WIMP masses\n are much smaller than\n the input 1$\\sigma$ value of \\mbox{40 km\/s}.\n\n In Table \\ref{tab:results_sh_Gau},\n we list the reconstructed values of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values of $v_0$.\n\n\\subsubsection{One--parameter shifted Maxwellian velocity distribution}\n\n\\plotGeSiGeshshvflat\n\n\\plotGeSiGeshshvGau\n\n In the previous Sec.~3.2.1,\n we have found that,\n by assuming (improperly)\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$,\n in both cases with and without an expectation value of\n the fitting parameter $v_0$,\n one would obtain a {\\em much higher} reconstruction result:\n \\mbox{$v_{\\rm 0, rec} \\simeq 295$ km\/s},\n which is 2$\\sigma$ to 4$\\sigma$ apart from\n the theoretical estimate of\n \\mbox{$v_0 \\approx 220$ km\/s}.\n This observation implies the need of\n a more suitable fitting velocity distribution function.\n Hence,\n as the second trial,\n we consider now the use of\n the {\\em shifted} Maxwellian velocity distribution $f_{1, \\sh}(v)$\n given in Eq.~(\\ref{eqn:f1v_sh})\n with {\\em only one} fitting parameter,\n i.e.~the Solar Galactic velocity $v_0$.\n In this case,\n we fix simply that\n\\beq\n   \\ve\n = 1.05\\~v_0\n\\~,\n\\label{eqn:ve_ave}\n\\eeq\n and neglect the time--dependence of $\\ve(t)$\n\\footnote{\n Note that\n hereafter we use $f_{1, \\sh, v_0}(v)$\n to denote the ``one--parameter''\n shifted Maxwellian velocity distribution function,\n in order to distinguish this\n from the ``original'' one\n given in Eq.~(\\ref{eqn:f1v_sh})\n with $v_0$ and $\\ve$\n as two independent fitting parameters.\n}\n\n In Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-flat},\n we consider first the flat probability distribution\n for the fitting parameter $v_0$\n with either the precisely known (input) (upper)\n or the reconstructed (lower) WIMP mass,\n respectively.\n It can be seen clearly that,\n with a more suitable assumption about\n the fitting function,\n one could indeed reconstruct the WIMP velocity distribution\n much closer to the true (input) one.\n Although\n no prior knowledge about $v_0$\n is used,\n this most important characteristic parameter\n could in principle be pinned down very precisely:\n the difference between the {\\em median} values of\n the reconstructed $v_0$\n and the true (input) one\n would be \\mbox{$\\lsim\\~{\\it 5}$ km\/s}\n (see also Table \\ref{tab:results_sh_sh_v0}).\n\n In addition,\n with the input WIMP mass,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the reconstructed $v_0$\n are only \\mbox{$^{+14.0}_{-11.2}\\~(^{+29.4}_{-23.8})$ km\/s}.\n Even with the reconstructed WIMP mass,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n could still be limited as small as only\n \\mbox{$^{+25.2}_{-22.4}\\~(^{+56.0}_{-44.8})$ km\/s}.\n It would be worth to emphasize that,\n compared to the uncertainty\n on the astronomical measurement of $v_0$\n of \\mbox{$\\sim 20$ km\/s} (or probably larger),\n the result offered by our Bayesian reconstruction method\n would be a pretty precise estimate\n and could help us to confirm\n the astronomical measurement of $v_0$.\n\n Moreover,\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau}\n we give the reconstruction results\n with the Gaussian probability distribution\n for $v_0$\n with an expectation value of \\mbox{$v_0 = 230$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{20 km\/s}.\n As summarized in Table \\ref{tab:results_sh_sh_v0},\n with a prior knowledge of the parameter $v_0$,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n could be reduced significantly\n to be \\mbox{$\\lsim$ 70\\%}.\n Remind here that\n the expectation value of the Gaussian probability distribution\n of $v_0$\n has been set as \\mbox{$v_0 = 230$ km\/s},\n a bit different from the input value.\n Nevertheless,\n our simulations show that\n this ``artificial'' (systematic) error\n could be corrected in our reconstruction process for $v_0$.\n\n In Table \\ref{tab:results_sh_sh_v0},\n we list the reconstructed results of $v_0$\n for all four considered cases\n with the one--parameter shifted Maxwellian velocity distribution\n $f_{1, \\sh, v_0}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values of $v_0$.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 217.4 & $217.4\\~^{+14.0}_{-11.2}\\~(^{+29.4}_{-23.8})$ & [206.2, 231.4] & [193.6, 246.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 221.6 & $221.6\\~^{+ 9.8}_{- 8.4}\\~(\\pm 18.2)$ & [213.2, 231.4] & [203.4, 239.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 218.8 & $218.8\\~^{+25.2}_{-22.4}\\~(^{+56.0}_{-44.8})$ & [196.4, 244.0] & [174.0, 274.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 221.6 & $221.6\\~^{+16.8}_{-15.4}\\~(\\pm 33.6)$ & [206.2, 238.4] & [188.0, 255.2] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the one--parameter shifted Maxwellian velocity distribution\n $f_{1, \\sh, v_0}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_sh_sh_v0}\n\n\\subsubsection{Shifted Maxwellian velocity distribution}\n\n\\plotGeshshGau\n\\plotGeSiGeshshGau\n\n Now we release the fixed relation between $v_0$ and $\\ve$\n given in Eq.~(\\ref{eqn:ve_ave})\n and consider the reconstruction of these two parameters\n {\\em simultaneously} and {\\em independently}.\n In addition,\n we assume here that,\n from the (naive) trials with the simple and one--parameter shifted\n Maxwellian velocity distributions\n done previously,\n one could already obtain a rough idea about\n the shape of the velocity distribution of halo WIMPs.\n This information would in turn give us\n the prior knowledge about\n the expectation values of the Solar and Earth's Galactic velocities\n $v_0$ and $\\ve$.\n Hence,\n we consider here only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\ve = 245$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n (see Table \\ref{tab:setup_sh}\n\\footnote{\n Remind that\n both of the expectation values of the fitting parameters $v_0$ and $\\ve$\n differ slightly from the true (input) values.\n Note also that\n the time--dependence of the Earth's Galactic velocity\n is ignored here\n and $\\ve$ is thus treated as\n a {\\em time--independent} fitting parameter.\n}.\n\n In Fig.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(a),\n we show the reconstructed\n shifted Maxwellian velocity distribution function\n as well as\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands.\n Here the true (input) WIMP mass has been used.\n Comparing to Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau}(a),\n it can be seen clearly that,\n with a prior knowledge about\n the Solar and Earth's Galactic velocities $v_0$ and $\\ve$,\n one could in principle reconstruct the velocity distribution function\n with two fitting parameters\n more precisely:\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n are much thinner and\n the deviations of $v_0$ and $\\ve$ are {\\em only a few} km\/s\n (see also Table \\ref{tab:results_sh_sh}\n\\footnote{\n Note however that,\n for using velocity distributions\n with two or more fitting parameters\n {\\em without} constraints on these parameters\n (i.e.~the use of the ``flat'' probability distribution),\n the distribution of the reconstructed results\n by our Bayesian analysis\n would be pretty wide,\n a part of them would even be\n on the boundary of the scanning ranges of these parameters.\n}.\n\n Fig.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(b) shows\n the distribution of\n the Bayesian reconstructed fitting parameter $v_0$ and $\\ve$\n in all simulated experiments\n on the $v_0 - \\ve$ plane.\n The light--green (light--blue, gold) points\n indicate the $1\\~(2)\\~(> 2)\\~\\sigma$ areas\n of the reconstructed combination of $v_0$ and $\\ve$.\n Note here that\n these $1\\~(2)\\~(> 2)\\~\\sigma$ areas are determined\n according to the {\\em descending} order of the ${\\rm p}_{\\rm median}$ values\n of the reconstructed combination of $v_0$ and $\\ve$.\n This means that\n the light--green (light--blue, gold) areas\n are the reconstructed combinations of $v_0$ and $\\ve$\n which give the largest 68.27\\% (95.45\\%) ${\\rm p}_{\\rm median}$ values\n in all of the simulated experiments.\n\n Moreover,\n the red upward--triangle indicates\n the input values of $v_0$ and $\\ve$,\n which has been labeled with the subscript ``gen''.\n The green disk shows\n the median values of the simulated results,\n whereas\n and blue downward--triangle\n the point which maximizes\n ${\\rm p}_{\\rm median}$.\n In addition,\n the thick (thin) green crosses show\n the $1\\~(2)\\~\\sigma$ (68.27\\% (95.45\\%)) ranges of the reconstructed results\n according to the order of\n the reconstructed values of $v_0$ or $\\ve$ {\\em along}\n (centered at their median values\n  $v_{0, {\\rm median}}$ or $v_{{\\rm e, median}}$).\n\n Additionally,\n in Figs.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(c) and (d),\n we give\n the distributions of\n the Bayesian reconstructed fitting parameters $v_0$ and $\\ve$\n as well as\n the $1\\~(2)\\~\\sigma$ statistical uncertainty ranges\n on the $v_0-$ and $\\ve-$ axes\n {\\em separately}.\n Note here that\n we project all reconstructed combinations of $\\abig{v_0, \\ve}$\n on the $v_0-$ or $\\ve-$axis.\n Comparing Fig.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(c)\n to Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau}(b),\n it can be found that,\n while\n the deviations of $v_0$ are a little bit larger\n then the results\n with the ``one--parameter'' fitting velocity distribution,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n are reduced to \\mbox{$\\sim$ 70\\%}\n (see Table \\ref{tab:results_sh_sh} and\n  Table \\ref{tab:results_sh_sh_v0}).\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 223.0 & $223.0 \\pm  7.0\\~(\\pm 14.0)$ & [216.0, 230.0] & [209.0, 237.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 223.0 & $223.0 \\pm 12.6\\~(^{+26.6}_{-23.8})$ & [210.4, 235.6] & [199.2, 249.6] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 238.4 & $238.4\\~^{+ 7.0}_{- 8.4}\\~(^{+15.4}_{-16.9})$ & [230.0, 245.4] & [221.6, 253.8] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 238.4 & $238.4 \\pm 12.6\\~(^{+25.2}_{-26.6})$ & [225.8, 251.0] & [211.8, 263.6] \\\\\n}\n{The reconstructed results of $v_0$ and $\\ve$\n with the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_sh_sh}\n\n As a comparison,\n in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}\n we use the reconstructed WIMP mass.\n Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}(b)\n shows that,\n as expected,\n the distribution of $v_0$ and $\\ve$ on the $v_0 - \\ve$ plane\n becomes wider and\n extends toward the directions of\n higher (lower)--$v_0$ and higher (lower)--$\\ve$.\n Nevertheless,\n the ``best--fit'' combination of them\n which maximizes ${\\rm p}_{\\rm median}$\n as well as\n the median values of $v_0$ and $\\ve$\n consistant very well\n with the true (input) values.\n\n In Table \\ref{tab:results_sh_sh_v0},\n we list the reconstructed results of $v_0$ and $\\ve$\n with the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n\n\\subsubsection{Variated shifted Maxwellian velocity distribution}\n\n\\plotGeshshDvGau\n\\plotGeSiGeshshDvGau\n\n In the previous Sec.~3.2.3,\n it has been found that,\n by using the shifted Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh})\n with two fitting parameters:\n $v_0$ and $\\ve$,\n one could reconstruct\n the ($1\\~(2)\\~\\sigma$ statistical uncertainty bands of the)\n velocity distribution function\n as well as\n pin down the Solar Galactic velocity $v_0$\n pretty precisely.\n However,\n as shown in Figs.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}(d)\n and \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}(d),\n the deviations of the reconstructed Earth's Galactic velocity $\\ve$\n from the true (input) value\n seem to be (much) larger than\n the deviations of the reconstructed $v_0$.\n This might be caused by\n the strong (anti--)correlation\n between $v_0$ and $\\ve$.\n Hence,\n we consider now a variation of\n the shifted Maxwellian distribution function,\n in the hope that\n this pretty large systematic deviation of $\\ve$\n could be reduced.\n\n We rewrite the shifted Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh})\n to the following ``variated'' form:\n\\beq\n   f_{1, \\sh, \\Delta v}(v)\n = \\frac{1}{\\sqrt{\\pi}} \\bfrac{v}{v_0 \\abrac{v_0 + \\Delta v}}\n   \\cbigg{  e^{-\\bbrac{v - \\abrac{v_0 + \\Delta v}}^2 \/ v_0^2}\n          - e^{-\\bbrac{v + \\abrac{v_0 + \\Delta v}}^2 \/ v_0^2}  }\n\\~,\n\\label{eqn:f1v_sh_Dv}\n\\eeq\n where\n\\beq\n        \\Delta v\n \\equiv \\ve - v_0\n\\label{eqn:Delta_v}\n\\eeq\n is the difference between $v_0$ and $\\ve(t)$\n\\footnote{\n As in the previous Sec.~3.2.3,\n the time--dependence of the Earth's Galactic velocity\n is ignored here\n and $\\Delta v$ is thus treated as\n a {\\em time--independent} fitting parameter.\n}\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 221.6 & $221.6 \\pm  8.4\\~(\\pm 15.4)$ & [213.2, 230.0] & [206.2, 237.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 221.6 & $221.6 \\pm 14.0\\~(\\pm 29.4)$ & [207.6, 235.6] & [192.2, 251.0] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n  11.1 & $ 11.1 \\pm  5.2\\~(^{+10.4}_{-11.7})$ & [5.9,  16.3] & [$- 0.6$,  21.5] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n  11.1 & $ 11.1 \\pm  7.8\\~(^{+15.6}_{-14.3})$ & [3.3,  18.9] & [$- 3.2$,  26.7] \\\\\n}\n{The reconstructed results of $v_0$ and $\\Delta v$\n with the variated shifted Maxwellian velocity distribution\n $f_{1, \\sh, \\Delta v}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_sh_sh_Dv}\n\n As in the previous Sec.~3.2.3,\n we assume here that,\n we have already a rough idea about\n the shape of the velocity distribution of halo WIMPs,\n and thus\n the prior knowledge about\n the expectation values of the (difference between the)\n Solar and Earth's Galactic velocities\n $v_0$ and $\\Delta v$.\n Hence,\n we consider here only the Gaussian probability distribution\n for both of the fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\Delta v = 15$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n (see Table \\ref{tab:setup_sh}).\n Two cases with both of\n the true (input) and the reconstructed WIMP masses\n have been considered.\n\n By comparing Figs.~\\ref{fig:f1v-Ge-100-0500-sh-sh_Dv-Gau}\n to Figs.~\\ref{fig:f1v-Ge-100-0500-sh-sh-Gau}\n and Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_Dv-Gau}\n to Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau},\n it can be seen obviously that\n the reconstructed velocity distribution function\n could indeed match the true (input) one much precisely,\n with however\n (slightly) wider $1\\~(2)\\~\\sigma$ statistical uncertainty bands.\n The systematic deviations of\n both fitting parameters\n from the true (input) values\n would also be much smaller than\n those given in the $v_0 - \\ve$ case.\n This implies importantly that,\n the use of our variated shifted velocity distribution function\n given in Eq.~(\\ref{eqn:f1v_sh_Dv})\n could indeed offer an estimate of\n the Earth's Galactic velocity $\\ve = v_0 + \\Delta v$\n with a much higher precision.\n Such a trick would be helpful to improve\n the estimation(s) of the Earth's Galactic velocity $\\ve$\n (and\/or other fitting parameter(s)).\n\n Note however that,\n as shown in Tables \\ref{tab:results_sh_sh}\n and \\ref{tab:results_sh_sh_Dv},\n by using the variated shifted Maxwellian velocity distribution,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the reconstructed $v_0$\n would be \\mbox{$\\sim$ 10\\%} larger.\n Meanwhile,\n from Eq.~(\\ref{eqn:Delta_v})\n the statistical uncertainty on $\\ve$\n can be estimated b\n\\footnote{\n Since,\n according to Eq.~(\\ref{eqn:Delta_v}),\n for a fixed value of $\\ve$\n two fitting parameters $v_0$ and $\\Delta v$\n should be ``anti--correlated''.\n}\n\\beqn\n      \\sigma\\abrac{\\ve}\n \\=   \\sqrt{  \\sigma^2\\abrac{v_0}\n            + \\sigma^2\\abrac{\\Delta v}\n            + 2 \\~ {\\rm cov}\\abrac{v_0, \\Delta v} \\sigma\\abrac{v_0} \\sigma\\abrac{\\Delta v}}\n      \\non\\\\\n \\eqnle  \\sqrt{  \\sigma^2\\abrac{v_0}\n            + \\sigma^2\\abrac{\\Delta v}}\n\\~.\n\\label{eqn:sigma_ve}\n\\eeqn\n Then,\n according to the results\n given in Table \\ref{tab:results_sh_sh_Dv},\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $\\ve$\n would be maximal \\mbox{$\\lsim~15\\%$} enlarged,\n whereas\n the systematic deviation of $\\ve$\n is much smaller.\n\n In Table \\ref{tab:results_sh_sh_Dv},\n we list the reconstructed results of $v_0$ and $\\Delta v$\n with the variated shifted Maxwellian velocity distribution\n $f_{1, \\sh, \\Delta v}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n\n\\subsection{Modified Maxwellian velocity distribution}\n\n In this subsection,\n we consider another recently often used theoretical\n one--dimensional WIMP velocity distribution function.\n In Refs.~\\cite{Lisanti10, Pato},\n a modification of the simple Maxwellian velocity distribution\n in Eq.~(\\ref{eqn:f1v_Gau})\n has been suggested:\n\\beq\n   f_{1, \\Gau, k}(v)\n = \\left\\{\\renewcommand{\\arraystretch}{1.5}\n          \\begin{array}{l c c}\n          \\frac{v^2}{N_{f, k}}\n          \\abrac{  e^{-v^2     \/ k v_0^2}\n                 - e^{-\\vmax^2 \/ k v_0^2}  }^k,\n          & ~~~~~~~~~~~~ &\n          ({\\rm for}~v \\le \\vmax), \\\\\n          0,\n          & ~~~~~~~~~~~~ &\n          ({\\rm for}~v >   \\vmax). \\\\\n          \\end{array}\n   \\right.\n\\label{eqn:f1v_Gau_k}\n\\eeq\n Here\n $\\vmax$ is the maximal cut-\u2013off\n velocity of $f_{1, \\Gau, k}(v)$\n and $N_{f, k}$ is the normalization constant\n depending on the value of the power index $k$.\n\n\\plotfvGauk\n\n\\begin{table}[b!]\n\\InsertSetupTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k}(v)$}\n{Simple\n & $v_0$      [km\/s] & $\\sim 220$ & [160,    300] &   230 & 20   \\\\\n\\hline\n 1--para. shifted\n & $v_0$      [km\/s] & $\\sim 175$ & [100,    300] &   200 & 40   \\\\\n\\hline\n \\multirow{2}{*}{Shifted}\n & $v_0$      [km\/s] & $\\sim 175$ & [100,    300] &   200 & 40   \\\\\n & $\\ve$      [km\/s] & $\\sim 185$ & [50,     300] &   200 & 40   \\\\\n\\hline\n \\multirow{2}{*}{Variated shifted}\n & $v_0$      [km\/s] & $\\sim 175$ & [100,    300] &   200 & 40   \\\\\n & $\\Delta v$ [km\/s] & $\\sim  10$ & [$-120$,  80] & $-20$ & 40   \\\\\n\\hline\n \\multirow{2}{*}{Modified}\n & $v_0$      [km\/s] &        220 & [160,    300] &   230 & 20   \\\\\n & $k$               &          2 & [0.5,    3.5] &     1 &  0.5 \\\\\n}\n{The input setup for\n the modified Maxwellian velocity distribution $f_{1, \\Gau, k}(v)$\n used for generating WIMP signals\n as well as\n the theoretically estimated values,\n the scanning ranges,\n the expectation values of\n and the 1$\\sigma$ uncertainties on\n the fitting parameters\n used for different fitting velocity distribution functions.\n}\n{tab:setup_Gau_k}\n\n In Fig.~\\ref{fig:f1v_Gau_k},\n we give\n the {\\em normalized}\n modified Maxwellian velocity distribution function $f_{1, \\Gau, k}(v)$\n with a common value of the Solar Galactic velocity\n \\mbox{$v_0 = 220$ km\/s}\n and different power indices $k$:\n $k = 1$ (dashed light--green),\n $k = 2$ (solid red),\n $k = 3$ (dash--dotted blue) and\n $k = 4$ (dotted magenta).\n As a comparison,\n the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n with \\mbox{$v_0 = 220$ km\/s}\n is also given\n as the short--dashed black curve.\n\n In Table \\ref{tab:setup_Gau_k},\n we list the input setup for\n the modified Maxwellian velocity distribution $f_{1, \\Gau, k}(v)$\n used for generating WIMP signals\n as well as\n the theoretically estimated values,\n the scanning ranges,\n the expectation values of\n and the 1$\\sigma$ uncertainties on\n the fitting parameters\n used for different fitting velocity distribution functions.\n\n\\subsubsection{Simple Maxwellian velocity distribution}\n\n\\plotGeSiGeGaukGauflat\n\n\\plotGeSiGeGaukGauGau\n\n As in Sec.~3.2,\n we start to fit to the reconstructed--input data points\n given by Eqs.~(\\ref{eqn:f1v_Qsn}) and (\\ref{eqn:cov_f1v_Qs_mu})\n with the simple Maxwellian velocity distribution function $f_{1, \\Gau}(v)$.\n\n In Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-flat},\n we use first the flat probability distribution\n for the fitting parameter $v_0$\n with either the precisely known (input) (upper)\n or the reconstructed (lower) WIMP mass,\n respectively.\n As shown in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau-Gau-flat}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau-Gau-flat},\n although\n prior knowledge about the Solar Galactic velocity is {\\em absent},\n one could in principle pin down the fitting parameter $v_0$ very precisely\n with systematic deviations of {\\em only a few} km\/s and\n $1\\~(2)\\~\\sigma$ statistical uncertainties of only\n \\mbox{$^{+11.2}_{-12.6}\\~(^{+36.4}_{-22.4})$ km\/s} and\n \\mbox{$^{+21.0}_{-18.2}\\~(^{+53.2}_{-32.2})$ km\/s}.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{simple   Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 217.4 & $217.4\\~^{+11.2}_{-12.6}\\~(^{+36.4}_{-22.4})$ & [204.8, 228.6] & [195.0, 253.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 218.8 & $220.2 \\pm  9.8\\~(^{+26.6}_{-18.2})$ & [210.4, 230.0] & [202.0, 246.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 218.8 & $217.4\\~^{+21.0}_{-18.2}\\~(^{+53.2}_{-32.2})$ & [199.2, 238.4] & [185.2, 270.6] \\\\\n \\cline{3-7}\n & & Gaussian &\n 220.2 & $220.2\\~^{+15.4}_{-14.0}\\~(^{+36.4}_{-26.6})$ & [206.2, 235.6] & [193.6, 256.6] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_k_Gau}\n\n In Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-Gau},\n we use the Gaussian probability distribution\n for $v_0$\n with an expectation value of \\mbox{$v_0 = 230$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{20 km\/s}.\n As summarized in Table \\ref{tab:results_Gau_k_Gau},\n the statistical uncertainties on the reconstructed $v_0$\n could now in principle be reduced by\n \\mbox{$\\sim 10$\\%} to \\mbox{$\\sim 30$\\%}.\n In addition,\n even with the reconstructed WIMP mass\n and thus a larger statistical fluctuation,\n the 1$\\sigma$ statistical uncertainties shown\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-Gau}\n are (much) smaller than the given uncertainty\n on the expectation value of $v_0$ (\\mbox{20 km\/s}).\n Moreover,\n although an expectation value of \\mbox{$v_0 = 230$ km\/s},\n which differs (slightly) from the true (input) one,\n is used,\n the value of the fitting parameter $v_0$\n could still be pinned down precisely\n with a negligible systematic deviation.\n\n Furthermore,\n remind here that,\n either,\n as shown in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-flat},\n two reconstructed\n (dashed green and dash-dotted blue)\n velocity distribution functions\n match the true (input) (solid red) one\n very well,\n but the reconstructed Solar Galactic velocity $v_0$\n would shift slightly away from the true (input) value,\n or,\n as shown in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-Gau},\n $v_0$ could be determined very precisely,\n but the reconstructed velocity distribution functions\n would differ slightly from the true (input) one.\n This is because of the fact that\n we have {\\em artificially} used an input value of $k = 2$\n and,\n as shown in Fig.~\\ref{fig:f1v_Gau_k},\n the modified Maxwellian velocity distribution function\n with $k = 2$\n is slightly different from the simple Maxwellian distribution.\n\n In Table \\ref{tab:results_Gau_k_Gau},\n we list the reconstructed results of $v_0$\n for all four considered cases\n with the simple Maxwellian velocity distribution $f_{1, \\Gau}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values of $v_0$.\n\n\\subsubsection{One--parameter shifted Maxwellian velocity distribution}\n\n\\plotGeSiGeGaukshvflat\n\n\\plotGeSiGeGaukshvGau\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 162.0 & $160.0\\~^{+10.0}_{- 8.0}\\~(^{+28.0}_{-14.0})$ & [152.0, 170.0] & [146.0, 188.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 162.0 & $162.0\\~^{+10.0}_{- 8.0}\\~(^{+26.0}_{-14.0})$ & [154.0, 172.0] & [148.0, 188.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 162.0 & $162.0 \\pm 14.0\\~(^{+36.0}_{-24.0})$ & [148.0, 176.0] & [138.0, 198.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 164.0 & $162.0\\~^{+16.0}_{-12.0}\\~(^{+38.0}_{-24.0})$ & [150.0, 178.0] & [138.0, 200.0] \\\\\n}\n{The reconstructed results of $v_0$\n for all four considered cases\n with the one--parameter shifted Maxwellian velocity distribution\n $f_{1, \\sh, v_0}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_k_sh_v0}\n\n Now,\n as a comparison of Sec.~3.3.1,\n we consider as the next trail the reconstruction with\n the one--parameter shifted Maxwellian velocity distribution function\n to fit the modified simple Maxwellian velocity distribution.\n\n As usual,\n in Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-flat}\n we use first the flat probability distribution\n for the fitting parameter $v_0$\n with either the precisely known (input) (upper)\n or the reconstructed (lower) WIMP mass,\n respectively.\n It can be seen that,\n firstly,\n although the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n could still cover the true (input) velocity distribution\n (somehow well),\n the systematic deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would be \\mbox{$\\sim 15$ km\/s}.\n The comparisons between reconstructed results shown in\n Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-flat}(a) and (c)\n to Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau-flat}(a) and (c)\n could indicate a high probability of the improper assumption of\n the fitting\n (one--parameter) shifted Maxwellian velocity distribution.\n\n Moreover,\n Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-flat}(b) and (d)\n (see also Table \\ref{tab:results_Gau_k_sh_v0})\n show clearly 4$\\sigma$ to even 6$\\sigma$ deviations of\n the reconstructed $v_0$\n from the true (input) value of \\mbox{$v_0 = 220$ km\/s}.\n As discussed in Sec.~3.2.1,\n this observation implies also\n an improper use of\n the (one--parameter) shifted Maxwellian velocity distribution\n as our fitting function.\n\n In Figs.\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-Gau},\n we use the Gaussian probability distribution\n for $v_0$\n with a {\\em slightly smaller} expectation value of \\mbox{$v_0 = {\\it 200}$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}.\n In contrast to our observations presented before,\n for this case\n the use of the Gaussian probability distribution\n would {\\em not} reduce the $1\\~(2)\\~\\sigma$ statistical uncertainties\n for both cases with the true (input) and\n the reconstructed WIMP masses\n (see Table \\ref{tab:results_Gau_k_sh_v0}).\n This would be caused by\n the large difference between\n the given expectation value\n and the (theoretically estimated) most suitable one of the parameter $v_0$\n (\\mbox{175 km\/s},\n  see Table \\ref{tab:setup_Gau_k}).\n This would indicate that,\n in practical use of our Bayesian reconstruction method,\n some trial--and--error tests\n for determining a more suitable expectation value of $v_0$\n (and the other fitting parameters)\n would be necessary and\n could then improve the fitting results.\n\n In Table \\ref{tab:results_Gau_k_sh_v0},\n we list the reconstructed results of $v_0$\n for all four considered cases\n with the one--parameter shifted Maxwellian velocity distribution\n $f_{1, \\sh, v_0}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values of $v_0$.\n\n\\subsubsection{Shifted Maxwellian velocity distribution}\n\n\\plotGeGaukshGau\n\\plotGeSiGeGaukshGau\n\n As in Sec.~3.2.3,\n we release now the fixed relation between $v_0$ and $\\ve$\n given in Eq.~(\\ref{eqn:ve_ave})\n and consider the reconstruction of these two parameters\n simultaneously and independently.\n In addition,\n we also assume here that,\n from the (naive) trials with the simple and one--parameter shifted\n Maxwellian velocity distributions\n done previously,\n one could already obtain a rough idea about\n the shape of the velocity distribution of halo WIMPs\n as well as\n expectation values of the Solar and Earth's Galactic velocities\n $v_0$ and $\\ve$.\n Hence,\n we consider here only the Gaussian probability distribution\n for both fitting parameters\n with a common expectation value of \\mbox{$v_0 = \\ve = {\\it 200}$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}\n (see Table \\ref{tab:setup_Gau_k}).\n Note also that,\n after some trial--and--error tests,\n we set the scanning ranges of $v_0$ and $\\ve$\n as \\mbox{$v_0 \\in [100, 300]$ km\/s} and\n \\mbox{$\\ve \\in [50, 300]$ km\/s}.\n\n In Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau},\n we consider first the case with the true (input) WIMP mass.\n As observed in Sec.~3.3.2,\n although the (low--velocity parts of the)\n $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n could cover the true (input) velocity distribution,\n the systematic deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would be \\mbox{$\\sim 10$ km\/s}\n (a little bit better then results shown\n  in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_v0-flat}(a) and (c)).\n Moreover,\n Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau}(c) and (d)\n show \\mbox{$\\gsim 4\\sigma$} deviations of\n the reconstructed values of $v_0$ and $\\ve$\n from the true (input, estimated) values of\n \\mbox{$v_0 = 220$ km\/s} and\n \\mbox{$\\ve = 231$ km\/s}.\n Additionally,\n the best--fit value of $\\ve$ is now even\n (\\mbox{$\\sim 10$ km\/s}) {\\em smaller} than\n the best--fit value of $v_0$.\n Hence,\n as discussed in Sec.~3.3.2,\n this observation\n (combined with the results given in Sec.~3.3.2)\n would indicate a high probability of\n the improper assumption of the fitting\n shifted Maxwellian velocity distribution.\n Moreover,\n in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh-Gau}\n we show our simulations with the reconstructed WIMP mass.\n The $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$ and $\\ve$\n would be \\mbox{$\\sim$ 10\\%} to \\mbox{$\\sim$ 50\\%} enlarged.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{shifted  Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 172.0 & $170.0 \\pm 12.0\\~(^{+38.0}_{-24.0})$ & [158.0, 182.0] & [146.0, 208.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 172.0 & $170.0\\~^{+18.0}_{-14.0}\\~(^{+44.0}_{-30.0})$ & [156.0, 188.0] & [140.0, 214.0] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 162.5 & $162.5\\~^{+17.5}_{-15.0}\\~(^{+42.5}_{-27.5})$ & [147.5, 180.0] & [135.0, 205.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 162.5 & $162.5\\~^{+20.0}_{-15.0}\\~(^{+48.1}_{-30.0})$ & [147.5, 182.5] & [132.5, 210.6] \\\\\n}\n{The reconstructed results of $v_0$ and $\\ve$\n with the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_k_sh}\n\n Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau} and\n \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh-Gau}\n as well as\n Table \\ref{tab:results_Gau_k_sh}\n show that\n one could fit an improper model\n to experimental data (somehow) well\n with combinations of special, probably unusual values of\n the fitting parameters.\n On one hand,\n the reconstructed results could offer us (rough) information\n about the (shape of the) velocity distribution of Galactic WIMPs.\n On the other hand,\n however,\n the observation that\n the reconstructed values of $v_0$ and $\\ve$\n are $2\\sigma$ to even $> 4\\sigma$ different\n from our (theoretically) expected values\n would indicate evidently the improper assumption about\n the fitting velocity distribution function.\n\n In Table \\ref{tab:results_Gau_k_sh},\n we list the reconstructed results of $v_0$ and $\\ve$\n with the shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n\n\\subsubsection{Variated shifted Maxwellian velocity distribution}\n\n\\plotGeGaukshDvGau\n\\plotGeSiGeGaukshDvGau\n\n As in Sec.~3.2.4,\n in order to correct\n the systematic deviations of the results given\n with the shifted Maxwellian velocity distribution function\n in Eq.~(\\ref{eqn:f1v_sh}),\n we consider here the use of\n its variation\n given in Eq.~(\\ref{eqn:f1v_sh_Dv}).\n As previously,\n we consider here only the Gaussian probability distribution\n for both fitting parameters $v_0$ and $\\Delta v$,\n with a common 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}.\n Moreover,\n according to the fitting results given in Sec.~3.3.3\n and some trial--and--error tests,\n we set \\mbox{$v_0 = {\\it 200}$ km\/s} and \\mbox{$\\Delta v = {\\it -20}$ km\/s}\n as the expectation values\n as well as\n \\mbox{$v_0 \\in [100, 300]$ km\/s} and\n \\mbox{$\\Delta v \\in [-120, 80]$ km\/s}\n as the scanning ranges.\n\n Comparing Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh_Dv-Gau}\n to Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau},\n it can be seen clearly that\n the use of the variated shifted Maxwellian velocity distribution\n could indeed offer a more reasonable and preciser reconstruction:\n not only that\n the ($1\\~(2)\\~\\sigma$ statistical uncertainty bands of the)\n reconstructed velocity distribution functions\n can match the true (input) one more closer,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$\n is also only \\mbox{$\\lsim$ 70\\%} of the uncertainties\n shown in Fig.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh-Gau}(c).\n Additionally,\n by using Eq.~(\\ref{eqn:sigma_ve}),\n the upper bound of the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on $\\ve$\n can be (approximately) given as:\n \\mbox{$^{+16.12}_{-12.80}\\~(^{+41.23}_{-24.41})$ km\/s},\n which is also maximal (approximately) equal to\n or even smaller than the uncertainties\n given in Table \\ref{tab:results_Gau_k_sh}.\n Hence,\n we would like to conclude that\n our introduction of the variated\n shifted Maxwellian velocity distribution\n (and\/or probably some other variations)\n could indeed be a useful trick\n for the practical use of our Bayesian reconstruction procedure.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 180.0 & $178.0 \\pm  8.0\\~(^{+26.0}_{-14.0})$ & [170.0, 186.0] & [164.0, 204.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 180.0 & $178.0\\~^{+14.0}_{-12.0}\\~(^{+34.5}_{-22.0})$ & [166.0, 192.0] & [156.0, 212.5] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n $-34.0$ & $-32.0\\~^{+14.0}_{-10.0}\\~(^{+32.0}_{-20.0})$ & [$-42.0$, $-18.0$] & [$-52.0$,   0.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n $-32.0$ & $-32.0\\~^{+14.0}_{-10.0}\\~(^{+30.5}_{-20.0})$ & [$-42.0$, $-18.0$] & [$-52.0$,  $- 1.5$] \\\\\n}\n{The reconstructed results of $v_0$ and $\\Delta v$\n with the variated shifted Maxwellian velocity distribution\n $f_{1, \\sh, \\Delta v}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n}\n{tab:results_Gau_k_sh_Dv}\n\n On the other hand,\n in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_Dv-Gau}\n we use the reconstructed WIMP mass.\n Comparing Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-sh_Dv-Gau}\n to Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-sh_Dv-Gau}\n (see also Table \\ref{tab:results_Gau_k_sh_Dv}),\n although\n the width of the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n of the reconstructed velocity distribution functions\n as well as\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$\n would be clearly larger,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $\\Delta v$\n are approximately equal to results\n with the true (input) WIMP mass.\n Meanwhile,\n even for this case with the reconstructed WIMP mass,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties on $v_0$ and $\\Delta v$\n are (much) smaller than the given 1$\\sigma$ uncertainty\n of their Gaussian probability distributions\n (\\mbox{40 km\/s}).\n\n Remind that,\n our simulations of the use of\n the variated shifted Maxwellian velocity distribution function\n for fitting data generated by (modified) simple Maxwellian one\n shown here\n indicates again that,\n by using an ``improper'' assumption about\n the fitting velocity distribution function\n with prior knowledge about\n the fitting parameters ($v_0$, $\\ve$ or $\\Delta v$),\n one would still reconstruct\n an approximate shape of the WIMP velocity distribution;\n the deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n could even be \\mbox{$\\lsim {\\it 10}$ km\/s},\n much smaller than the commonly used 1$\\sigma$ uncertainty\n on the Solar Galactic velocity of \\mbox{$\\sim 20$ km\/s}.\n\n However,\n our simulations show also that,\n with an ``improper'' assumption about\n the fitting velocity distribution,\n one would obtain an ``unexpected'' result\n about each single fitting parameter.\n E.g.~here we get 3$\\sigma$ to 5$\\sigma$ deviations\n of the reconstructed Solar Galactic velocity\n from the theoretical estimate\n (see\n  Table\n  \\ref{tab:results_Gau_k_sh_Dv})\n and large ``negative'' values for\n the difference between the Solar and Earth's Galactic velocities.\n This observation indicates clearly that\n our initial assumption about\n the fitting velocity distribution function\n would be incorrect \n or at least need to be modified.\n\n In Table \\ref{tab:results_Gau_k_sh_Dv},\n we list the reconstructed results of $v_0$ and $\\Delta v$\n with the variated shifted Maxwellian velocity distribution\n $f_{1, \\sh, \\Delta v}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n\n\\subsubsection{Modified Maxwellian velocity distribution}\n\n\\plotGeGaukGaukGauflat\n\\plotGeSiGeGaukGaukGauflat\n\n As the last tested fitting velocity distribution function\n with data generated by the modified Maxwellian velocity distribution\n given by Eq.~(\\ref{eqn:f1v_Gau_k}),\n we consider now the reconstruction of\n the modified Maxwellian velocity distribution itself\n with two fitting parameters:\n the Solar Galactic velocity $v_0$ and\n the power index $k$.\n\n In Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-Gau_k-Gau-flat}\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau_k-Gau-flat},\n we use the Gaussian probability distribution\n for the fitting parameter $v_0$\n with an expectation value of \\mbox{$v_0 = 230$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n but\n the flat distribution\n for the parameter $k$,\n with either the precisely known (input)\n or the reconstructed WIMP mass,\n respectively.\n Remind that,\n in Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-Gau_k-Gau-flat}(d)\n and \\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau_k-Gau-flat}(d)\n the bins at $k = 0.5$ and $k = 3.5$ are ``overflow'' bins.\n This means that\n they contain also the experiments\n whose best--fit values of $k$ would be\n either $k < 0.5$ or $k > 3.5$.\n\n First,\n as shown in Sec.~3.3.1,\n the Solar Galactic velocity $v_0$\n could be pinned down very precisely:\n a small systematic deviation of \\mbox{$<$ 10 km\/s}\n and 1$\\sigma$ statistical uncertainty of\n \\mbox{$\\lsim\\~10$ km\/s}\n could be achieved.\n However,\n Figs.~\\ref{fig:f1v-Ge-100-0500-Gau_k-Gau_k-Gau-flat}(b) and (d)\n as well as\n Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-Gau_k-Gau_k-Gau-flat}(b) and (d)\n show that,\n due to the small difference between\n the modified Maxwellian velocity distribution\n with different power indices $k$\n (see Fig.~\\ref{fig:f1v_Gau_k})\n and the large statistical fluctuation and\n 1$\\sigma$ reconstruction uncertainty\n with only 500 WIMP events (on average),\n our Bayesian reconstruction of the velocity distribution\n would be {\\em totally non--sensitive}\n on the second fitting parameter (power index) $k$.\n\n Nevertheless,\n the wide spread of the reconstructed power index $k$\n (in particular,\n  the high cumulative numbers in the (overflow) bin $k = 3.5$)\n and,\n in contrast,\n the narrow widths of\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands of\n the reconstructed velocity distribution functions\n imply that,\n for reconstructing the rough information about\n the one--dimensional WIMP velocity distribution,\n a precise value of the power index $k$\n would {\\em not be crucial},\n and\n the {\\em simple} Maxwellian velocity distribution\n $f_{1, \\Gau}(v)$ given in Eq.~(\\ref{eqn:f1v_Gau})\n would already be a good approximatio\n\\footnote{\n As described in Ref.~\\cite{Lisanti10},\n the modification of the simple Maxwellian velocity distribution function\n $f_{1, \\Gau, k}(v)$ given in Eq.~(\\ref{eqn:f1v_Gau_k})\n has significant difference from\n the simple Maxwellian one given in Eq.~(\\ref{eqn:f1v_Gau})\n in the high--velocity tail.\n Moreover,\n in Refs.~\\cite{YYMao, Kuhlen13},\n another empirical modification of\n the simple Maxwellian velocity distribution\n with also a significant difference\n in the high--velocity tail\n has been introduced.\n Unfortunately,\n our simulations presented here\n show that\n it would be {\\em impossible}\n to distinguish these subtle modifications\n by our Bayesian reconstruction method\n (with only a few hundreds of recorded WIMP events).\n\n On the other hand,\n simulations with other (well--motivated) halo models\n and different fitting velocity distribution functions\n can be tested on the {\\tt AMIDAS} website\n \\cite{AMIDAS-web,\n       AMIDAS-II}.\n}.\n\n In Table \\ref{tab:results_Gau_k_Gau_k},\n we give\n the reconstructed results of $v_0$ and $k$\n with the modified Maxwellian velocity distribution\n $f_{1, \\Gau, k}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n Note that,\n since the 1$\\sigma$ lower and upper bounds of the median values of $k$\n are already beyond our scanning range,\n the 2$\\sigma$ bounds are meanless to give here.\n\n\\begin{table}[b!]\n\\InsertResultsTable\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k = 2}(v)$}\n{modified Maxwellian velocity distribution $f_{1, \\Gau, k}(v)$}\n{\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 228.6 & $227.2\\~^{+ 7.0}_{- 8.4}\\~(^{+19.6}_{-18.2})$ & [218.8, 234.2] & [209.0, 246.8] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 227.2 & $227.2\\~^{+ 9.8}_{-12.6}\\~(^{+29.4}_{-28.0})$ & [214.6, 237.0] & [199.2, 256.6] \\\\\n\\hline\n \\multirow{2}{*}{$k$} &\n Input &\n Flat &\n 3.47 & $3.38                  $ & [0.5, 3.5] & $\\times$ \\\\\n \\cline{2-7}\n & Reconst. &\n Flat &\n 2.72 & $3.29                  $ & [0.5, 3.5] & $\\times$ \\\\\n}\n{The reconstructed results of $v_0$ and $k$\n with the modified Maxwellian velocity distribution\n $f_{1, \\Gau, k}(v)$\n as well as\n the $1\\~(2)\\~\\sigma$ uncertainty ranges\n of the median values.\n Note that,\n firstly,\n we use here the Gaussian probability distribution for $v_0$\n but the flat one for $k$.\n Secondly,\n since the 1$\\sigma$ lower and upper bounds of the median values of $k$\n are already beyond our scanning range,\n the 2$\\sigma$ bounds are meanless to give here.\n}\n{tab:results_Gau_k_Gau_k}\n\n\\subsection{For different WIMP masses}\n\n\\plotGeSiGeshGauflatL\n\n\\plotGeSiGeshshGauL\n\n\\plotGeSiGeshshDvGauL\n\n In the previous Secs.~3.1 to 3.3,\n we fixed the input WIMP mass as \\mbox{$\\mchi = 100$ GeV}.\n As a further test of our Bayesian reconstruction method\n for the one--dimensional WIMP velocity distribution,\n in this subsection,\n we consider the cases for\n either a light WIMP mass of \\mbox{$\\mchi = 25$ GeV}\n or a heavy WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n\n Here we consider\n only the {\\em shifted} Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh})\n for generating WIMP events;\n three fitting functions:\n simple,\n shifted\n and variated shifted\n Maxwellian velocity distributions\n will be tested.\n All input setup and fitting parameters\n are the same as in Sec.~3.2\n (see Table \\ref{tab:setup_sh}).\n Additionally,\n only the {\\em reconstructed} WIMP mass is used.\n\n\\subsubsection{For a light WIMP mass}\n\n We consider first\n a rather light WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n Note that,\n firstly,\n since for our tested targets $\\rmXA{Si}{28}$ and $\\rmXA{Ge}{76}$,\n the kinematic maximal cut--off energies\n given in Eq.~(\\ref{eqn:Qmax_kin})\n are only \\mbox{$Q_{\\rm max, kin, Si} = 68.12$ keV}\n and \\mbox{$Q_{\\rm max, kin, Ge} = 52.65$ keV},\n respectively,\n the maximal experimental cut--off energy for both targets\n in our simulations demonstrated here\n has been reduced to only \\mbox{$\\Qmax = 50$ keV}.\n Secondly,\n since the lighter the WIMP mass,\n the shaper the expected recoil energy spectrum,\n the width of the first energy bin in Eq.~(\\ref{eqn:bn_delta})\n has been set as \\mbox{$b_1 = 5$ keV}\n and the total number of bins\n has been reduced to only {\\em four} bins\n between $\\Qmin$ and $\\Qmax$ ($B = 4$);\n up to {\\em two} bins have been combined to a window\n and thus {\\em four} windows ($W = 4$)\n will be reconstructe\n\\footnote{\n The last window is neglected automatically\n in the \\amidas\\ code,\n due to a very few expected event number\n in the last bin (window).\n}$^{,~}\n\\footnote{\n It has been found that,\n by reducing the total number of the energy bins\n (and in turn that of the windows)\n and thus collecting more events in one bin (window),\n the Bayesian reconstructed velocity distribution\n could be improved (significantly).\n}.\n\n\\paragraph{Simple Maxwellian velocity distribution \\\\}\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 296.8 & $294.4\\~^{+14.4}_{-12.0}\\~(^{+31.2}_{-21.6})$ & [282.4, 308.8] & [272.8, 325.6] \\\\\n \\cline{3-7}\n & & Gaussian &\n 294.4 & $294.4\\~^{+ 9.6}_{-12.0}\\~(^{+24.0}_{-21.6})$ & [282.4, 304.0] & [272.8, 318.4] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 292.0 & $289.6\\~^{+28.8}_{-24.0}\\~(^{+62.4}_{-43.2})$ & [265.6, 318.4] & [246.4, 352.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 292.0 & $289.6 \\pm 21.6\\~(^{+48.0}_{-40.8})$ & [268.0, 311.2] & [248.8, 337.6] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 220.2 & $218.8\\~^{+ 9.8}_{- 7.0}\\~(^{+19.6}_{-14.0})$ & [211.8, 228.6] & [204.8, 238.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 221.6 & $221.6 \\pm  7.0\\~(^{+15.4}_{-14.0})$ & [214.6, 228.6] & [207.6, 237.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 217.4 & $214.6\\~^{+21.0}_{-16.8}\\~(^{+44.8}_{-30.8})$ & [197.8, 235.6] & [183.8, 259.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 218.8 & $218.8 \\pm 15.4\\~(^{+32.2}_{-29.4})$ & [203.4, 234.2] & [189.4, 251.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 221.6 & $221.6 \\pm  7.0\\~(^{+15.4}_{-12.9})$ & [214.6, 228.6] & [208.7, 237.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 218.8 & $220.2\\~^{+12.6}_{-14.0}\\~(^{+28.0}_{-26.6})$ & [206.2, 232.8] & [193.6, 248.2] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 237.0 & $237.0\\~^{+ 7.0}_{- 8.4}\\~(^{+14.0}_{-16.8})$ & [228.6, 244.0] & [220.2, 251.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 234.2 & $234.2 \\pm 12.6(\\pm 25.2)$ & [221.6, 246.8] & [209.0, 259.4] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 221.6 & $221.6 \\pm  5.6\\~(^{+12.6}_{-11.2})$ & [216.0, 227.2] & [210.4, 234.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 218.8 & $218.8\\~^{+14.0}_{-12.6}\\~(^{+29.4}_{-23.8})$ & [206.2, 232.8] & [195.0, 248.2] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n   9.8 & $ 11.1\\~^{+ 3.9}_{- 5.2}\\~(^{+ 9.1}_{-11.7})$ & [5.9,  15.0] & [$- 0.6$,  20.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n   9.8 & $  9.8 \\pm  6.5\\~(^{+14.3}_{-13.0})$ & [3.3,  16.3] & [$- 3.2$,  24.1] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for the input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n}\n{tab:results_sh_L}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-Gau-flat}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(c) and (d)),\n we use first the (improper)\n simple Maxwellian velocity distribution function\n with one parameter $v_0$\n to fit the reconstructed--input data.\n As the first trial of the reconstruction of\n the one--dimensional WIMP velocity distribution\n {\\em without} prior knowledge about the Solar Galactic velocity,\n the flat probability distribution\n has been used here\n (results with the Gaussian probability distribution\n  are given in Table \\ref{tab:results_sh_L}).\n\n Although we use the improper assumption about\n the fitting velocity distribution\n and have only {\\em four} available data points\n (solid black vertical bars),\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n could still cover the true (input) velocity distribution\n with a systematic deviation of the peak of\n the reconstructed velocity distribution\n of \\mbox{$\\lsim$ 15 km\/s}\n from that of the true (input) one.\n However,\n the best--fit values of the Solar Galactic velocity\n are now \\mbox{$\\simeq 294$ km\/s}\n and \\mbox{$\\sim 3\\sigma$} apart from the theoretically expected value.\n\n Moreover,\n our further simulations\n with the Gaussian probability distribution\n for the fitting parameter $v_0$\n with an expectation value of \\mbox{$v_0 = {\\it 280}$ km\/s}\n and a 1$\\sigma$ uncertainty of \\mbox{{\\em 40} km\/s}\n show that,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n for such a light WIMP mass\n could be reduced to \\mbox{$\\sim 60$\\%} to \\mbox{$\\sim 80$\\%}\n (compare Table \\ref{tab:results_sh_L} to Table \\ref{tab:results_sh_Gau}).\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh-Gau}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}),\n the shifted Maxwellian velocity distribution\n with two fitting parameters $v_0$ and $\\ve$\n has been tested to fit to the reconstructed--input data.\n Only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\ve = 245$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n is considered here.\n\n Astonishingly,\n with {\\em only four} available data points\n the reconstructed velocity distribution functions\n could match the true (input) one very precisely:\n the systematic deviation of $v_0$ is negligible\n and that of $\\ve$ is {\\em only a few} km\/s,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on two fitting parameters are also only\n \\mbox{$^{+12.6}_{-14.0}\\~(^{+28.0}_{-26.6})$} and\n \\mbox{$\\pm 12.6(\\pm 25.2)$},\n respectively.\n\n\\paragraph{Variated shifted Maxwellian velocity distribution \\\\}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh_Dv-Gau}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_Dv-Gau}),\n the variated shifted Maxwellian velocity distribution\n with two parameters $v_0$ and $\\Delta v$\n has been tested to fit to the reconstructed--input data.\n Only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\Delta v = 15$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n is considered here.\n\n It can be seen that,\n although\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n are a bit wider than\n those given with the shifted Maxwellian distribution,\n with only four available data points\n the reconstructed velocity distribution function\n could also match the true (input) one very well.\n Moreover,\n as shown in Secs.~3.2.4 and 3.3.4,\n the second fitting parameter $\\Delta v$\n could also be pinned down very precisely\n with a negligible systematic deviation\n (see Table \\ref{tab:results_sh_L}).\n\n ~\n\n In Table \\ref{tab:results_sh_L},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for the input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n Both cases with\n the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\subsubsection{For a heavy WIMP mass}\n\n\\plotGeSiGeshGauflatH\n\n\\plotGeSiGeshshGauH\n\n\\plotGeSiGeshshDvGauH\n\n We consider now\n a rather heavy WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n The maximal experimental cut--off energy for both targets\n in our simulations demonstrated here\n has been set again as \\mbox{$\\Qmax = 100$ keV}.\n And,\n as usual,\n the width of the first energy bin in Eq.~(\\ref{eqn:bn_delta})\n has been set as \\mbox{$b_1 = 10$ keV},\n {\\em five} bins between $\\Qmin$ and $\\Qmax$ are used ($B = 5$)\n and up to {\\em three} bins have been combined to a window ($W = 6$).\n\n\\paragraph{Simple Maxwellian velocity distribution \\\\}\n\n For a heavy WIMP mass,\n due to the (large) statistical fluctuation\n discussed in Ref.~\\cite{DMDDmchi},\n our reconstructed velocity distribution functions\n given in Fig.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-Gau-flat}(a)\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-Gau-flat}(c)\n  and \\ref{fig:f1v-Ge-SiGe-025-0500-sh-Gau-flat}(a))\n have clearly\n a (much) wider $1\\~(2)\\~\\sigma$ statistical uncertainty bands;\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the reconstructed Solar Galactic velocity\n are also (much) larger as\n \\mbox{$^{+33.6}_{-31.2}\\~(^{+64.8}_{-60.0})$ km\/s}.\n And,\n as shown in Fig.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-Gau-flat}(b),\n a considerable fraction of the reconstructed $v_0$\n would excess our scanning upper bound of \\mbox{400 km\/s}.\n Remind that\n the bin at \\mbox{$v_0 = 400$ km\/s}\n is an ``overflow'' bin,\n which contains also the experiments\n with the best--fit $v_0$ value of \\mbox{$> 400$ km\/s}.\n\n Nevertheless,\n comparing to the much larger 1$\\sigma$ statistical uncertainty\n on the reconstructed--input data (solid black vertical bars),\n our Bayesian reconstruction with\n an in fact improper fitting velocity distribution\n could still offer an approximation\n with only \\mbox{$\\lsim\\~15$ km\/s}\n systematic deviation of the peak of\n the reconstructed velocity distribution\n from that of the true (input) one.\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh-Gau}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau}\n  and \\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh-Gau}),\n the shifted Maxwellian velocity distribution\n with two fitting parameters $v_0$ and $\\ve$\n has been tested to fit to the reconstructed--input data.\n Only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\ve = 245$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n is considered here.\n\n With a proper fitting velocity distribution\n and {\\em one more} fitting parameter,\n the reconstructed velocity distribution\n could now match the true (input) one very precisely\n with much narrower $1\\~(2)\\~\\sigma$ statistical uncertainty bands.\n Additionally,\n the $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the fitting parameters $v_0$ and $\\ve$\n can be significantly reduced to only\n \\mbox{$\\pm 9.8\\~(^{+22.4}_{-18.2})$} and\n \\mbox{$^{+14.0}_{-11.2}\\~(^{+26.6}_{-21.0})$},\n respectively.\n Note that,\n as shown in Table \\ref{tab:results_sh_H},\n the use of the {\\em one--parameter}\n shifted Maxwellian velocity distribution\n with {\\em only one} fitting parameter $v_0$\n and the fixed relation between $v_0$ and $\\ve$\n would give\n a (much) wider $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n of the reconstructed velocity distribution\n as well as\n a (much) larger $1\\~(2)\\~\\sigma$ statistical uncertainties\n on the reconstructed Solar Galactic velocity:\n \\mbox{$\\sim 50$\\%} to a factor of \\mbox{$\\sim 2$} larger.\n\n\\paragraph{Variated shifted Maxwellian velocity distribution \\\\}\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 287.2 & $287.2\\~^{+28.8}_{-21.6}\\~(^{+69.6}_{-40.8})$ & [265.6, 316.0] & [246.4, 356.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 284.8 & $284.8\\~^{+19.2}_{-16.8}\\~(^{+36.0}_{-31.2})$ & [268.0, 304.0] & [253.6, 320.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 289.6 & $289.6\\~^{+55.2}_{-43.2}\\~(^{+110.4}_{-74.4})$ & [246.4, 344.8] & [215.2, 400.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 289.6 & $287.2\\~^{+33.6}_{-31.2}\\~(^{+64.8}_{-60.0})$ & [256.0, 320.8] & [227.2, 352.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 209.0 & $209.0\\~^{+18.2}_{-15.4}\\~(^{+43.4}_{-26.6})$ & [193.6, 227.2] & [182.4, 252.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 218.8 & $218.8 \\pm  9.8\\~(\\pm 19.6)$ & [209.0, 228.6] & [199.2, 238.4] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 211.8 & $210.4\\~^{+37.8}_{-29.4}\\~(^{+89.6}_{-50.4})$ & [181.0, 248.2] & [160.0, 300.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 218.8 & $220.2 \\pm 18.2\\~(^{+36.4}_{-35.0})$ & [202.0, 238.4] & [185.2, 256.6] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 223.0 & $223.0\\~^{+ 5.6}_{- 7.0}\\~(^{+12.6}_{-14.0})$ & [216.0, 228.6] & [209.0, 235.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 223.0 & $224.4 \\pm  9.8\\~(^{+22.4}_{-18.2})$ & [214.6, 234.2] & [206.2, 246.8] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 237.0 & $237.0 \\pm  7.0\\~(^{+14.0}_{-15.4})$ & [230.0, 244.0] & [221.6, 251.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 237.0 & $237.0\\~^{+14.0}_{-11.2}\\~(^{+26.6}_{-21.0})$ & [225.8, 251.0] & [216.0, 263.6] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 218.8 & $218.8\\~^{+ 9.8}_{- 8.4}\\~(\\pm 18.2)$ & [210.4, 228.6] & [200.6, 237.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 220.2 & $220.2\\~^{+15.4}_{-16.8}\\~(^{+32.2}_{-30.8})$ & [203.4, 235.6] & [189.4, 252.4] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n   9.8 & $  9.8 \\pm  5.2\\~(^{+ 9.1}_{-10.4})$ & [4.6,  15.0] & [$- 0.6$,  18.9] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n   9.8 & $  9.8\\~^{+ 9.1}_{- 7.8}\\~(^{+18.2}_{-15.6})$ & [2.0,  18.9] & [$- 5.8$,  28.0] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for the input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n}\n{tab:results_sh_H}\n\n Finally,\n in Figs.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh_Dv-Gau}\n (cf.~Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_Dv-Gau}\n  and \\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh_Dv-Gau}),\n we test the possibility of\n improving the reconstruction precision\n by the use of\n the variated shifted Maxwellian velocity distribution\n with two fitting parameters $v_0$ and $\\Delta v$.\n Only the Gaussian probability distribution\n for both fitting parameters\n with expectation values of \\mbox{$v_0 = 230$ km\/s}\n and \\mbox{$\\Delta v = 15$ km\/s}\n and a common 1$\\sigma$ uncertainty of \\mbox{20 km\/s}\n is considered here.\n\n While\n the reconstructed velocity distribution function\n could still match the true (input) one very precisely\n with however\n wider $1\\~(2)\\~\\sigma$ statistical uncertainty bands,\n the ``best--fit'' values of\n both parameters $v_0$ and $\\Delta v$\n (and in turn $\\ve$)\n could again be very precisely determined\n with negligible systematic deviations\n (see Table \\ref{tab:results_sh_H}).\n\n ~\n\n Note that,\n since the heavier the WIMP mass,\n the smaller the transformation constant $\\alpha$\n defined in Eq.~(\\ref{eqn:alpha}),\n for an experimental maximal cut--off energy\n \\mbox{$\\Qmax \\approx 100$ GeV},\n the reconstructible velocity range\n of our model--independent data analysis method\n would be much smaller than our maximal cut--off velocity $\\vmax$\n (e.g.~\\mbox{$\\sim 285$ km\/s}\n  for \\mbox{$\\mchi = 250$ GeV}\n  and the $\\rmXA{Ge}{76}$ target).\n Therefore,\n our simulations shown in this subsection\n demonstrate meaningfully that,\n our Bayesian reconstruction of\n the one--dimensional WIMP velocity distribution\n would be an important improvement\n for offering more and preciser information\n about the Galactic halo,\n e.g.~the position of the peak of the WIMP velocity distribution,\n for the WIMP mass between \\mbox{$\\cal O$(20) GeV}\n and even \\mbox{$\\cal O$(500) GeV}.\n\n In Table \\ref{tab:results_sh_H},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for the input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n Both cases with the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\subsection{Background effects}\n\n\\plotdRdQbg\n\n In this last part of\n our presentation of the numerical simulations\n of the Bayesian reconstruction of\n the WIMP velocity distribution function,\n we consider the effects of {\\em unrejected} background events.\n Similar to our earlier works\n in Refs.~\\cite{DMDDbg-mchi, DMDDbg-f1v},\n we take into account a small fraction of\n {\\em artificially} generated background events\n in the fake experimental data sets\n and want to study\n how well the WIMP velocity distribution\n as well as\n the fitting parameters\n could be reconstructed.\n\n In all simulations demonstrated in this subsection,\n a combination of\n the {\\em target--dependent exponential} form\n of the residue background spectrum\n introduced in Ref.~\\cite{DMDDbg-mchi}\n with a small {\\em constant} component\n has been used:\n\\beq\n    \\aDd{R}{Q}_{\\rm bg}\n =  \\aDd{R}{Q}_{\\rm bg, ex}\n  + r_{\\rm const} \\aDd{R}{Q}_{\\rm bg, const}\n\\~,\n\\label{eqn:dRdQ_bg}\n\\eeq\n where\n\\beq\n   \\aDd{R}{Q}_{\\rm bg, ex}\n = \\exp\\abrac{-\\frac{Q \/{\\rm keV}}{A^{0.6}}}\n\\~,\n\\label{eqn:dRdQ_bg_ex}\n\\eeq\n and\n\\beq\n   \\aDd{R}{Q}_{\\rm bg, const}\n = 1\n\\~.\n\\label{eqn:dRdQ_bg_const}\n\\eeq\n Here $Q$ is the recoil energy,\n $A$ is the atomic mass number of the target nucleus.\n The power index of $A$, 0.6, is an empirical constant,\n which has been chosen so that\n the exponential background spectrum is\n somehow similar to,\n but still different from\n the expected recoil spectrum of the target nuclei;\n otherwise,\n there is in practice no difference between\n the WIMP scattering and background spectr\n\\footnote{\n Note that,\n among different possible choices,\n we use in our simulations the atomic mass number $A$\n as the simplest, unique characteristic parameter\n in the general analytic form (\\ref{eqn:dRdQ_bg_ex})\n for defining the artificial residue background spectra\n for different target nuclei.\n However,\n it does not mean that\n the (superposition of the real) background spectra\n would depend simply\/primarily on $A$ or\n on the mass of the target nucleus, $\\mN$.\n In other words,\n it is practically equivalent to\n use the expression (\\ref{eqn:dRdQ_bg_ex})\n or $(dR \/ dQ)_{\\rm bg, ex} = e^{-Q \/ 13.5~{\\rm keV}}$ directly\n for a $\\rmXA{Ge}{76}$ target\n (cf.~\\cite{Green-mchi08}).\n}.\n Additionally,\n $r_{\\rm const}$ is the ratio\n between the exponential and constant components\n in the total {\\em background} spectrum,\n which has been fixed as $r_{\\rm const} = 0.05$\n for all simulations.\n\n Note that,\n firstly,\n as argued in Ref.~\\cite{DMDDbg-mchi},\n the exponential form of background spectrum\n is rather naive;\n but,\n since we consider here\n only {\\em a few tens residue} background events\n induced by {\\em several different} sources,\n pass all discrimination criteria,\n and then mix with other WIMP--induced events\n in our data sets of a few hundreds {\\em total} events,\n an exact form of background spectrum\n for each target nucleus\n would not be crucial and\n the exponential + constant form of background spectrum\n in Eq.~(\\ref{eqn:dRdQ_bg})\n should practically not be unrealistic.\n Secondly,\n as demonstrated in Refs.~\\cite{DMDDf1v, DMDDmchi}\n and in the previous subsections,\n our Bayesian reconstruction of\n the one--dimensional WIMP velocity distribution\n requires only measured recoil energies\n and occasionally prior knowledge about\n the Solar and Earth's Galactic velocities.\n Hence,\n for applying this method to future real experimental data,\n prior knowledge about (different) background source(s)\n is {\\em not required at all}.\n\n In Figs.~\\ref{fig:dRdQ-bg-ex-const-000-100-20-100},\n we show\n the measured recoil energy spectrum (solid red histogram)\n for a $\\rmXA{Ge}{76}$ (a) and a $\\rmXA{Si}{28}$ (b) targets\n with an input WIMP mass of \\mbox{$\\mchi = 100$ GeV}.\n The dotted blue curve is\n the elastic WIMP--nucleus scattering spectrum\n for {\\em generating} signal events,\n whereas\n the dashed green curve shows\n the {\\em artificial} background spectrum:\n the exponential background spectrum\n given in Eq.~(\\ref{eqn:dRdQ_bg_ex})\n accompanied with an extra constant component,\n normalized to fit to the background ratio of 20\\%.\n\n\\plotGeSiGeshshvGaubg\n\n\\plotGeSiGeshshGaubg\n\n\\subsubsection{For a moderate WIMP mass}\n\n We consider first a moderate input WIMP mass of\n \\mbox{$\\mchi = 100$ GeV}.\n The {\\em shifted} Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh})\n is used for generating WIMP signals.\n All input setup and fitting parameters\n are the same as in Sec.~3.2\n (see Table \\ref{tab:setup_sh})\n and\n a fraction of 20\\% background events\n has been taken into account.\n Additionally,\n as in Sec.~3.4,\n we consider only the use of\n the Gaussian probability distribution\n for the fitting parameters:\n $v_0$ and $\\ve$ or $\\Delta v$\n as well as\n the use of the reconstructed WIMP mass.\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 284.8 & $284.8\\~^{+24.0}_{-19.2}\\~(^{+52.8}_{-40.8})$ & [265.6, 308.8] & [244.0, 337.6] \\\\\n \\cline{3-7}\n & & Gaussian &\n 284.8 & $284.8 \\pm 16.8\\~(^{+36.0}_{-33.6})$ & [268.0, 301.6] & [251.2, 320.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 253.6 & $253.6\\~^{+36.0}_{-31.2}\\~(^{+79.2}_{-60.0})$ & [222.4, 289.6] & [193.6, 332.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 258.4 & $258.4\\~^{+28.8}_{-26.4}\\~(^{+57.6}_{-55.2})$ & [232.0, 287.2] & [203.2, 316.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 211.8 & $211.8\\~^{+16.8}_{-15.4}\\~(^{+36.4}_{-30.8})$ & [196.4, 228.6] & [181.0, 248.2] \\\\\n \\cline{3-7}\n & & Gaussian &\n 217.4 & $218.8\\~^{+ 9.8}_{-11.2}\\~(^{+21.0}_{-23.8})$ & [207.6, 228.6] & [195.0, 239.8] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 188.0 & $188.0\\~^{+26.6}_{-22.4}\\~(^{+56.0}_{-28.0})$ & [165.6, 214.6] & [160.0, 244.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 202.0 & $202.0\\~^{+18.2}_{-16.8}\\~(^{+36.4}_{-35.0})$ & [185.2, 220.2] & [167.0, 238.4] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 225.8 & $225.8\\~^{+ 7.0}_{- 8.4}\\~(\\pm 16.8)$ & [217.4, 232.8] & [189.4, 239.8] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 214.6 & $214.6\\~^{+12.2}_{-12.6}\\~(\\pm 25.2)$ & [202.0, 226.8] & [189.4, 239.8] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 231.4 & $231.4 \\pm 8.4\\~(^{+16.8}_{-18.2})$ & [223.0, 239.8] & [213.2, 248.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 220.2 & $220.2\\~^{+12.6}_{-11.2}\\~(^{+25.2}_{-23.8})$ & [209.0, 232.8] & [196.4, 245.4] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 220.2 & $220.2\\~^{+ 8.4}_{- 9.8}\\~(\\pm 19.6)$ & [210.4, 228.6] & [200.6, 239.8] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 204.8 & $204.8\\~^{+16.8}_{-15.4}\\~(^{+32.2}_{-30.8})$ & [189.4, 221.6] & [174.0, 237.0] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n   5.9 & $  5.9\\~^{+ 5.2}_{- 6.5}\\~(^{+10.4}_{-13.0})$ & [$- 0.6$,  11.1] & [$- 7.1$,  16.3] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n $- 0.6$ & $- 0.6 \\pm  7.8\\~(\\pm 15.6)$ & [$- 8.4$,   7.2] & [$-16.2$,  15.0] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for data sets mixed with 20\\% background events\n and the input WIMP mass of \\mbox{$\\mchi = 100$ GeV}.\n}\n{tab:results_sh_bg}\n\n\\paragraph{One--parameter shifted Maxwellian velocity distribution \\\\}\n\n In Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a),\n it can be seen first that,\n due to the extra background events\n in both of the low and high energy ranges\n (see Figs.~\\ref{fig:dRdQ-bg-ex-const-000-100-20-100}),\n the reconstructed--input data (solid black vertical bars)\n would be shifted (strongly) to the low--velocity rang\n\\footnote{\n Note that,\n as shown in Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a)\n and \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau-bg}(a),\n the reconstructed WIMP mass\n is now {\\em overestimated}:\n \\mbox{$m_{\\chi, {\\rm rec}} \\approx 136$ GeV}.\n}:\n the peak of the solid black crosses is now at \\mbox{$\\sim 220$ km\/s},\n i.e.~\\mbox{$\\sim 90$ km\/s} smaller then\n the position of the true (input) velocity distribution.\n However,\n our simulation shown in\n Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a)\n indicates clearly and importantly that,\n by assuming the shifted Maxwellian WIMP velocity distribution\n and the {\\em time--averaged} relation\n between the Solar and Earth's Galactic velocities,\n the reconstructed WIMP velocity distributions\n could alleviate this systematic shift:\n the deviations of the peaks of\n the ($1\\~(2)\\~\\sigma$ statistical uncertainty bands of the)\n reconstructed velocity distributions\n would only be \\mbox{$\\sim\\~30\\~^{+30}_{-20}\\~(^{+60}_{-40})$ km\/s}.\n\n In fact,\n it has also been found that,\n once an (approximately) precisely determined (true) WIMP mass\n could be used,\n the reconstructed WIMP velocity distribution\n could match the true (input) one very precisely:\n the deviation of the reconstructed $v_0$\n would be \\mbox{$\\lsim\\~10$ km\/s} (flat)\n or even negligible (Gaussian)\n (see Table \\ref{tab:results_sh_bg}).\n\n Note that,\n although a fraction of 20\\% unrejected background events\n has been mixed (artificially) into the analyzed (pseudo--)data sets,\n the (1$\\sigma$ statistical uncertainty on the)\n median value of the reconstructed $v_0$'s\n (\\mbox{$202.0\\~^{+18.2}_{-16.8}$ km\/s})\n would still cover the true (input) Solar Galactic velocity of\n \\mbox{$v_0 = 220$ km\/s}.\n Moreover,\n once we take into account the statistical fluctuation\n of the reconstructed--input data (the solid black vertical bars),\n the effect of 20\\% residue background events\n on reconstructing information about\n the (shape of the) WIMP velocity distribution function\n would {\\em not be very significant}.\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau-bg},\n we release the fixed relation between $v_0$ and $\\ve$\n and determine these two parameters\n simultaneously and independently.\n\n It can be seen that,\n firstly,\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n are obviously narrower than those shown\n in Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a);\n the deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would be reduced to only \\mbox{$\\lsim\\~15$ km\/s}.\n In addition,\n the systematic deviations and\n the (1$\\sigma$ statistical uncertainties on the)\n median values of the reconstructed fitting paramaters $v_0$ and $\\ve$\n shown in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau-bg}(c) and (d)\n are also (much) smaller than that shown in\n Fig.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(b)\n (see Table \\ref{tab:results_sh_bg}).\n Note here that,\n as given in Table \\ref{tab:results_sh_bg},\n once an (approximately) precisely determined (true) WIMP mass\n could be used,\n one could reconstruct\n the WIMP velocity distribution function\n very precisely\n with very small or even negligible systematic deviations\n of both two fitting parameters.\n This means that,\n our Bayesian reconstruction method\n for the WIMP velocity distribution function\n would {\\em not be affected (significantly)}\n by a fraction of \\mbox{$\\sim 20$\\%}\n unrejected background events\n mixed in the analyzed data sets\n (for a WIMP mass of ${\\cal O}(100)$ GeV). \n\n ~\n\n In Table \\ref{tab:results_sh_bg},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for data sets mixed with 20\\% background events\n and the input WIMP mass of \\mbox{$\\mchi = 100$ GeV}.\n Both cases with the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\subsubsection{For a light WIMP mass}\n\n Now,\n we consider the case\n with a light input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n Simulation setup is the same as in Sec.~3.4.1\n and\n a fraction of 20\\% background events\n has been taken into account.\n\n\\plotGeSiGeshshvGaubgL\n\n\\plotGeSiGeshshGaubgL\n\n\\paragraph{One--parameter shifted Maxwellian velocity distribution \\\\}\n\n Since the reconstructed WIMP mass\n would be \\mbox{$\\sim 30$\\%} {\\em overestimated}\n (\\mbox{$m_{\\chi, {\\rm rec}} \\approx 31.5$ GeV})\n due to the extra background events\n and {\\em only four} reconstructed--input data points are available,\n Fig.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh_v0-Gau-bg}(a)\n shows that\n the ``best--fit'' {\\em one--parameter}\n shifted Maxwellian velocity distribution functions\n would match\n not the true (input) velocity distribution (solid red curve),\n but the analyzed data points (solid black crosses)\n well.\n Nevertheless,\n at least,\n the 2$\\sigma$ statistical uncertainty band\n of the reconstructed velocity distributions\n could cover the true (input) one;\n the systematic deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would also only be \\mbox{$\\sim 30$ km\/s}.\n Meanwhile,\n the 2$\\sigma$ statistical uncertainty on the\n median value of the reconstructed $v_0$'s\n (\\mbox{$200.6^{+29.4}_{-26.6}$ km\/s})\n would still cover\n the true (input) Solar Galactic velocity of\n \\mbox{$v_0 = 220$ km\/s}.\n This could be further improved\n by using an (approximately) precisely determined (true) WIMP mass\n to be\n \\mbox{$227.2\\~^{+ 9.8}_{- 8.4}$ km\/s} (flat) and\n \\mbox{$228.6 \\pm  7.0$ km\/s} (Gaussian)\n (see Table \\ref{tab:results_sh_bg_L}).\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n As the case of the \\mbox{100 GeV} WIMP mass\n shown in Figs.~\\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh_v0-Gau-bg}(a)\n and \\ref{fig:f1v-Ge-SiGe-100-0500-sh-sh-Gau-bg}(a),\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n reconstructed with the shifted Maxwellian velocity distribution\n with two independent fitting parameters $v_0$ and $\\ve$\n shown in Fig.~\\ref{fig:f1v-Ge-SiGe-025-0500-sh-sh-Gau-bg}(a)\n would clearly be much narrower than those\n reconstructed with only one parameter $v_0$.\n Meanwhile,\n in contrast to other (presented) cases,\n our simulations with the {\\em true} ({\\em input}) WIMP mass\n show that\n the reconstructed--input data\n as well as\n the ($1\\~(2)\\~\\sigma$ statistical uncertainty bands of the)\n reconstructed velocity distribution function\n would slightly shift to the {\\em high}--velocity range.\n\n ~\n\n Furthermore,\n comparing results given in Table \\ref{tab:results_sh_bg_L}\n to those in Table \\ref{tab:results_sh_bg},\n it has been found interesting and probably importantly that,\n for an (input) WIMP mass of \\mbox{$\\cal O$(20) GeV},\n the use of our {\\em variated} shifted Maxwellian velocity distribution\n given in Eq.~(\\ref{eqn:f1v_sh_Dv})\n could offer preciser reconstruction results\n with (relatively) smaller statistical uncertainties,\n although\n fewer (four in our simulations) data points are available.\n\n In Table \\ref{tab:results_sh_bg_L},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for data sets mixed with 20\\% background events\n and the input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n Both cases with the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 308.8 & $306.4\\~^{+16.8}_{-12.0}\\~(^{+36.0}_{-24.0})$ & [294.4, 323.2] & [282.4, 342.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 304.0 & $304.0 \\pm 12.0\\~(\\pm 24.0)$ & [292.0, 316.0] & [280.0, 328.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 260.8 & $260.8\\~^{+24.0}_{-21.6}\\~(^{+51.0}_{-40.8})$ & [239.2, 284.8] & [220.0, 311.8] \\\\\n \\cline{3-7}\n & & Gaussian &\n 265.6 & $263.2\\~^{+21.6}_{-19.2}\\~(^{+43.2}_{-38.4})$ & [244.0, 284.8] & [224.8, 306.4] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 228.6 & $227.2\\~^{+ 9.8}_{- 8.4}\\~(^{+22.4}_{-15.4})$ & [218.8, 237.0] & [211.8, 249.6] \\\\\n \\cline{3-7}\n & & Gaussian &\n 228.6 & $228.6 \\pm  7.0\\~(^{+15.4}_{-14.0})$ & [221.6, 235.6] & [214.6, 244.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 193.6 & $193.6\\~^{+16.8}_{-15.4}\\~(^{+35.4}_{-29.4})$ & [178.2, 210.4] & [164.2, 229.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 200.6 & $200.6\\~^{+15.4}_{-14.0}\\~(^{+29.4}_{-26.6})$ & [186.6, 216.0] & [174.0, 230.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 230.0 & $230.0 \\pm  7.0\\~(\\pm 14.0)$ & [223.0, 237.0] & [216.0, 244.0] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 207.6 & $207.6 \\pm 12.6\\~(^{+23.8}_{-25.2})$ & [195.0, 220.2] & [182.4, 231.4] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 239.8 & $239.8\\~^{+ 8.4}_{- 7.0}\\~(\\pm 15.4)$ & [232.8, 248.2] & [224.4, 255.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 218.8 & $220.2\\~^{+11.2}_{-12.6}\\~(^{+23.8}_{-25.2})$ & [207.6, 231.4] & [195.0, 244.0] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 228.6 & $228.6 \\pm  7.0\\~(^{+14.0}_{-12.6})$ & [221.6, 235.6] & [216.0, 242.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 203.4 & $204.8 \\pm 12.6\\~(\\pm 23.8)$ & [192.2, 217.4] & [181.0, 228.6] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n  12.4 & $ 11.1 \\pm  5.2\\~(\\pm 10.4)$ & [5.9,  16.3] & [0.7,  21.5] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n   2.0 & $  2.0 \\pm  6.5\\~(\\pm 13.0)$ & [$- 4.5$,   8.5] & [$-11.0$,  15.0] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for data sets mixed with 20\\% background events\n and the input WIMP mass of \\mbox{$\\mchi = 25$ GeV}.\n}\n{tab:results_sh_bg_L}\n\n\\subsubsection{For a heavy WIMP mass}\n\n As the last case,\n we consider here\n a heavy input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n Simulation setup is the same as in Sec.~3.4.2.\n Note however that,\n since the constant component of the background spectrum\n used in our simulations\n would cause a strongly {\\em overestimated} WIMP mass,\n in particular,\n once WIMPs are heavy\n (e.g.~the \\mbox{250 GeV} input WIMP mass\n  would now be reconstructed as \\mbox{$\\simeq 338$ GeV})\n \\cite{DMDDbg-mchi},\n the ratio of the background events\n in the analyzed data sets\n has been set as {\\em only 10\\%}\n \\cite{DMDDbg-f1v}.\n\n\\paragraph{One--parameter shifted Maxwellian velocity distribution \\\\}\n\n As usual,\n we consider first the one--parameter\n shifted Maxwellian velocity distribution function\n to fit the reconstructed--input data points.\n\n\\plotGeSiGeshshvGaubgH\n\n\\plotGeSiGeshshGaubgH\n\n In Figs.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh_v0-Gau-bg},\n we can see unexpectedly that,\n although the input WIMP mass is pretty heavy,\n the systematic deviations of the peak positions of\n the reconstructed WIMP velocity distribution functions\n from that of the true (input) one\n would be only \\mbox{$\\sim 10$ km\/s}\n (for 10\\% background ratio!).\n This might be due to that,\n as shown in Sec.~3.4.2,\n for our used experimental maximal cut--off energy\n \\mbox{$\\Qmax = 100$ GeV}\n and the $\\rmXA{Ge}{76}$ target,\n the reconstructible velocity range\n would only be\n \\mbox{$\\sim 270$ km\/s}\n (shifted slightly to the {\\em low}--velocity range\n  due to the {\\em overestimated} WIMP mass)\n and thus\n this maximal reconstructible velocity\n is theoretically smaller than\n the position of the peak of the velocity distribution function\n (see e.g.~Fig.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh_v0-Gau-bg}(a)).\n This means that\n the approximately {\\em monotonically increased} shape of\n the reconstructed--input data points (solid black vertical bars)\n with pretty large 1$\\sigma$ statistical uncertainties\n would alleviate the effects of the overestimations of\n the analyzed (reconstructed--input) data points\n and the reconstructed WIMP mass\n caused by the extra background events.\n\n\\begin{table}[t!]\n\\InsertResultsTable\n{shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$}\n{simple  Maxwellian velocity distribution $f_{1, \\Gau}(v)$}\n{\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 277.6 & $277.6\\~^{+26.4}_{-21.6}\\~(^{+64.8}_{-38.4})$ & [256.0, 304.0] & [239.2, 342.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 277.6 & $278.8 \\pm 18.0\\~(^{+37.2}_{-32.4})$ & [260.8, 296.8] & [246.4, 316.0] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 263.2 & $263.2\\~^{+48.0}_{-38.4}\\~(^{+115.2}_{-62.4})$ & [224.8, 311.2] & [200.8, 378.4] \\\\\n \\cline{3-7}\n & & Gaussian &\n 268.0 & $268.0\\~^{+33.6}_{-31.2}\\~(^{+67.2}_{-55.2})$ & [236.8, 301.6] & [212.8, 335.2] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  one--parameter shifted Maxwellian velocity distribution $f_{1, \\sh, v_0}(v)$} \\\\\n\\hline\n \\multirow{4}{*}{$v_0$ [km\/s]} &\n \\multirow{2}{*}{Input} &\n Flat &\n 203.4 & $203.4\\~^{+19.2}_{-14.0}\\~(^{+44.8}_{-28.0})$ & [189.4, 222.6] & [175.4, 248.2] \\\\\n \\cline{3-7}\n & & Gaussian &\n 216.0 & $216.0\\~^{+ 9.8}_{-11.2}\\~(^{+19.6}_{-21.0})$ & [204.8, 225.8] & [195.0, 235.6] \\\\\n \\cline{2-7}\n & \\multirow{2}{*}{Reconst.} &\n Flat &\n 195.0 & $193.6\\~^{+33.6}_{-28.0}\\~(^{+78.4}_{-33.6})$ & [165.6, 227.2] & [160.0, 272.0] \\\\\n \\cline{3-7}\n & & Gaussian &\n 211.8 & $210.4 \\pm 18.2\\~(^{+36.4}_{-35.0})$ & [192.2, 228.6] & [175.4, 246.8] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  shifted Maxwellian velocity distribution $f_{1, \\sh}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 221.6 & $223.0\\~^{+ 5.6}_{- 8.4}\\~(^{+12.6}_{-15.4})$ & [214.6, 228.6] & [207.6, 235.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 218.8 & $220.2\\~^{+ 9.8}_{- 8.4}\\~(^{+22.4}_{-18.2})$ & [211.8, 230.0] & [202.0, 242.6] \\\\\n\\hline\n \\multirow{2}{*}{$\\ve$ [km\/s]} &\n Input &\n Gaussian &\n 232.8 & $234.2\\~^{+ 7.0}_{- 8.4}\\~(^{+14.0}_{-16.8})$ & [225.8, 241.2] & [217.4, 248.2] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 230.0 & $230.0\\~^{+12.6}_{- 9.8}\\~(^{+25.2}_{-21.0})$ & [220.2, 242.6] & [209.0, 255.2] \\\\\n\\hline\n\\hline\n \\multicolumn{7}{|| c ||}\n {\\bf\\boldmath\n  Reconstruction:\n  variated shifted Maxwellian velocity distribution $f_{1, \\sh, \\Delta v}(v)$} \\\\\n\\hline\n \\multirow{2}{*}{$v_0$ [km\/s]} &\n Input &\n Gaussian &\n 216.0 & $217.4\\~^{+ 8.4}_{- 9.8}\\~(^{+18.2}_{-19.6})$ & [207.6, 225.8] & [197.8, 235.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n 210.4 & $211.8\\~^{+16.8}_{-15.4}\\~(^{+33.6}_{-30.8})$ & [196.4, 228.6] & [181.0, 245.4] \\\\\n\\hline\n \\multirow{2}{*}{$\\Delta v$ [km\/s]} &\n Input &\n Gaussian &\n   7.2 & $  7.2\\~^{+ 5.2}_{- 5.2}\\~(^{+10.4}_{-11.7})$ & [2.0,  12.4] & [$- 4.5$,  17.6] \\\\\n \\cline{2-7}\n & Reconst. &\n Gaussian &\n   4.6 & $  4.6\\~^{+ 7.8}_{- 7.8}\\~(^{+16.9}_{-14.3})$ & [$- 3.2$,  12.4] & [$- 9.7$,  21.5] \\\\\n}\n{The reconstructed results\n with four fitting velocity distribution functions\n for data sets mixed with {\\em 10\\%} background events\n and the input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n}\n{tab:results_sh_bg_H}\n\n\\paragraph{Shifted Maxwellian velocity distribution \\\\}\n\n Now\n we use the shifted Maxwellian velocity distribution function\n with two independent fitting parameters $v_0$ and $\\ve$\n to fit the reconstructed--input data points.\n Astonishingly and unexpectedly\n (probably accidentally),\n Figs.~\\ref{fig:f1v-Ge-SiGe-250-0500-sh-sh-Gau-bg}\n show that\n both of the ``best--fit'' results of the parameters $v_0$ and $\\ve$\n are {\\em almost exact} as the true (input) values\n and\n the 1$\\sigma$ statistical uncertainties on $v_0$ and $\\ve$\n are only \\mbox{$\\sim 10$ km\/s}.\n\n Meanwhile,\n in contrast to our simulation results\n with the variated shifted Maxwellian velocity distribution function\n shown previously,\n for the case of the 250 GeV WIMP mass\n with 10\\% background ratio,\n the Bayesian reconstructed parameter $\\Delta v$\n could have a (much) larger deviations\n from the true (estimated) value\n (see Table \\ref{tab:results_sh_bg_H})!\n\n ~\n\n In Table \\ref{tab:results_sh_bg_H},\n we give the reconstructed results\n with all four fitting velocity distribution functions\n for data sets mixed with {\\em 10\\%} background events\n and the input WIMP mass of \\mbox{$\\mchi = 250$ GeV}.\n Both cases with the true (input) and the reconstructed WIMP masses\n have been simulated and summarized.\n\n\\section{Summary and conclusions}\n\n In this paper,\n we extended our earlier work on\n the development of the model--independent data analysis method\n for the reconstruction of the (time--averaged) one--dimensional\n velocity distribution of Galactic WIMPs\n and introduced the Bayesian fitting procedure\n of the theoretical velocity distribution functions.\n\n In this fitting procedure,\n the (rough) velocity distribution\n reconstructed by using raw experimental data,\n i.e.~measured recoil energies,\n with one or more different target nuclei\n has been used as reconstructed--input data (points).\n By assuming a fitting WIMP velocity distribution function\n and scanning the parameter space\n based on the Bayesian analysis,\n the (fitting) astronomical characteristic parameters,\n e.g.~the Solar and Earth's Galactic velocities $v_0$ and $\\ve$,\n would be pinned down as the output results\n and thus\n the functional form of the one--dimensional velocity distribution\n can be reconstructed\n (instead of only a few discrete points).\n\n As the first test of our Bayesian reconstruction method\n for the one--dimensional WIMP velocity distribution function,\n we used the simplest isothermal spherical Galactic halo model\n for both generating WIMP--signal events\n and as the assumed velocity distribution\n with the unique fitting parameter:\n the Solar Galactic velocity $v_0$.\n Our simulations show that,\n with (only) 500 recorded events (on average)\n and without prior knowledge about the Solar Galactic velocity,\n $v_0$ could in principle be pinned down\n with a negligible deviation and\n a 1$\\sigma$ statistical uncertainty of only \\mbox{$\\sim 12$ km\/s}\n (with a precisely known WIMP mass)\n or \\mbox{$\\sim 20$ km\/s}\n (with a reconstructed WIMP mass),\n respectively.\n Moreover,\n once (rough) information about the Solar Galactic velocity\n can be given,\n the statistical uncertainties on the reconstructed $v_0$\n could even be reduced to \\mbox{$\\sim 70$\\%}.\n \n For more realistic consideration,\n we then took into account\n the orbital motion of the Solar system around our Galaxy\n as well as\n that of the Earth around the Sun\n and\n turned to use the shifted Maxwellian velocity distribution function\n for generating WIMP signals.\n As comparisons,\n four different fitting\n functions\n have been considered:\n the simple and\n the (one--parameter and variated) shifted Maxwellian velocity distributions.\n It has been found\n that,\n firstly,\n with an improper assumed fitting function\n (e.g.~the simple Maxwellian velocity distribution here),\n the WIMP velocity distribution\n could still be reconstructed and\n offer some important information about Galactic WIMPs,\n e.g.~the rough position of the peak of\n the one--dimensional velocity distribution function.\n The deviations of the peaks of\n the reconstructed velocity distributions\n from that of the true (input) one\n would be only \\mbox{$\\sim 10$ km\/s}.\n However,\n the best--fit value(s) of the fitting parameter(s)\n would be unexpected\/unreasonable.\n For instance,\n the reconstructed Solar Galactic velocity $v_0$\n would be\n 2$\\sigma$ (with the reconstructed WIMP mass)\n or even\n 4$\\sigma$ (with the input WIMP mass)\n apart from its theoretical estimate.\n Such an observation\n could in turn be an important criterion\n on the assumption of fitting velocity distribution function.\n\n Moreover,\n our simulations with\n the (one--parameter and variated)\n shifted Maxwellian velocity distributions\n show that,\n although in all of these three cases\n the reconstructed velocity distributions\n could match the true (input) one pretty precisely,\n with two fitting parameters\n the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n of the reconstructed velocity distributions\n would be narrower then those\n with only one fitting parameter.\n In addition,\n the use of the variation of the shifted Maxwellian velocity distribution\n could (strongly) reduce the systematic deviations of\n the determinations of the characteristic\n Solar and Earth's Galactic velocities $v_0$ and $\\ve$,\n with however a bit larger statistical uncertainties.\n\n Furthermore,\n we considered also a modification of\n the simple Maxwellian velocity distribution\n with an extra power index\n as the generating WIMP velocity distribution.\n First,\n we used the simple Maxwellian velocity distribution\n without the power index\n as the test fitting function.\n Since the difference between the modification and the original\n simple Maxwellian velocity distributions\n are very tiny,\n the reconstructed velocity distribution function\n could match the true (input) one very precisely\n and the characteristic Solar Galactic velocity\n could also be reconstructed\n with negligible systematic deviation.\n\n Meanwhile,\n our simulations with the\n (one--parameter and variated)\n shifted Maxwellian velocity distribution functions\n show that,\n although the positions of the peak of\n the reconstructed velocity distribution\n would be only \\mbox{$\\lsim\\~10$ km\/s} deviated\n from the true (input) one,\n a clear 2$\\sigma$ to even 6$\\sigma$ difference\n between the best--fit values of\n the Solar and\/or the Earth's Galactic velocities\n and the true (input) ones\n could be observed.\n Such results would in turn\n indicate evidently the improper assumption of\n the shifted Maxwellian velocity distribution function.\n\n Moreover,\n from the simulations with\n the modified simple Maxwellian velocity distribution\n with the power index\n as the second fitting parameter,\n it has been found that\n our Bayesian reconstruction of the WIMP velocity distribution\n would be (totally) non--sensitive\n on the power index.\n This means that,\n unfortunately,\n with only a few hundreds of recorded WIMP events\n it would still be impossible\n to distinguish (evidently)\n different subtle variations of\n the (simple and shifted) WIMP velocity distribution functions.\n\n As comparisons,\n we considered also a light and a heavy input WIMP masses.\n For the case of light WIMPs,\n due to the sharp shapes of the recoil energy spectra\n and the small kinetic maximal cut--off energies,\n the recorded WIMP events\n would need to be separated into fewer bins\/windows.\n However,\n our simulations show that,\n with only four available reconstructed--input data points,\n the true (input) velocity distribution function\n could astonishingly be reconstructed very precisely.\n On the other hand,\n once WIMPs are heavy,\n the statistical fluctuation on the reconstructed WIMP mass\n becomes pretty large\n and hence\n the Bayesian reconstructions of the velocity distribution\n as well as\n of the Solar and Earth's Galactic velocities\n would have large statistical uncertainties.\n Nevertheless,\n the reconstructed velocity distribution function\n with the best--fit characteristic Solar and Earth's Galactic velocities\n could still match the true (input) one very precisely.\n\n Finally,\n the effects of residue (unrejected) background events\n mixed in data sets to analyze\n have also been considered.\n Three different WIMP masses\n with background ratios of 10\\% or 20\\%\n have been tested.\n Although,\n due to the choice of our artificial residue background spectrum,\n the reconstructed WIMP masses\n would be overestimated\n and the (rough shape of) the reconstructed--input data points\n would thus be shifted (significantly) to lower velocities,\n the functional forms of the chosen fitting velocity distributions\n could somehow alleviate these systematic shifts\n and the $1\\~(2)\\~\\sigma$ statistical uncertainty bands\n could still cover the true (input) velocity distribution.\n In particular,\n for heavy WIMPs,\n since the reconstructed--input data points\n should be in the velocity range smaller than\n the position of the peak of the velocity distribution function,\n its approximately monotonically increased shape\n with pretty large 1$\\sigma$ statistical uncertainties\n would alleviate the effects of the overestimations of\n the analyzed (reconstructed--input) data points\n and the reconstructed WIMP mass\n caused by the extra background events.\n The reconstructed velocity distribution function\n could then match the true (input) one pretty well.\n\n It would be worth to emphasize that,\n first,\n comparing to the pretty large\n (1$\\sigma$) statistical uncertainties\n on the reconstructed--input data points\n (offered by our model--independent method\n  developed in Ref.~\\cite{DMDDf1v}\n  with raw experimental measured recoil energies),\n our Bayesian reconstruction of\n the WIMP velocity distribution function\n introduced here\n with only a few km\/s deviation and\n \\mbox{$\\cal O$(10) km\/s} 1$\\sigma$ statistical uncertainties\n on the reconstructed Solar and Earth's Galactic velocities\n would be a remarkable improvement.\n\n Second,\n all our simulations show importantly that,\n even initial values different slightly from the true (input) setup\n have been used as the expectation values\n for the Gaussian probability distribution of the fitting parameters,\n these fitting parameters\n could still be pinned down (pretty) precisely.\n As long as\n a proper assumed fitting velocity distribution function\n is used,\n the best--fit values of the reconstructed parameters\n could always be less than 1$\\sigma$\n apart from the true (input\/theoretical) values.\n This observation indicates that\n rough, slightly incorrect prior knowledge\n about our fitting parameters\n would not affect (significantly)\n the reconstructed results\n in our Bayesian reconstruction procedure.\n\n Moreover,\n by rewriting the functional form of\n the (basic) fitting velocity distribution function,\n one could not only pin down the fitting parameters more precisely,\n but also occasionally reduce the statistical uncertainties\n on the reconstructed parameters.\n\n In summary,\n we developed in this paper\n the Bayesian reconstruction procedure\n for fitting theoretically predicted models of\n the one--dimensional WIMP velocity distribution function\n to data (points),\n which can be reconstructed directly\n from experimental measured recoil energies.\n Hopefully,\n this extension of our earlier work\n could offer more useful information about the Dark Matter halo,\n which could further be used\n in e.g.~indirect DM detection experiments.\n\n\\subsubsection*{Acknowledgments}\n The author appreciates Mei-Yu Wang\n for useful discussions\n about models of the velocity distribution of Galactic WIMPs.\n The author would also like to thank\n the Physikalisches Institut der Universit\\\"at T\\\"ubingen\n for the technical support of the computational work\n presented in this paper\n as well as\n the friendly hospitality of\n the Graduate School of Science and Engineering for Research,\n University of Toyama,\n the Institute of Modern Physics,\n Chinese Academy of Sciences,\n the Center for High Energy Physics,\n Peking University,\n and\n the Xinjiang Astronomical Observatory,\n Chinese Academy of Sciences,\n where part of this work was completed.\n This work\n was partially supported\n by the National Science Council of R.O.C.\n under the contracts no.~NSC-98-2811-M-006-044 and\n no.~NSC-99-2811-M-006-031\n as well as\n the CAS Fellowship for Taiwan Youth Visiting Scholars\n under the grant no.~2013TW2JA0002.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe giant dipole resonance (GDR) is the best-known fundamental mode of nuclear excitations at high frequencies. The GDR built on the ground state of heavy nuclei has a small width ($\\sim$ 4 - 5 MeV) and the integrated cross section up to around 30 MeV that exhausts the Thomas-Reich-Kuhn (TRK) sum rule. The GDR built on highly excited compound (CN) nuclei was observed for the first time in 1981~\\cite{Newton}, and at present a wealth of experimental data has been accumulated for the GDR widths at finite temperature $T$ and angular momentum $J$ in various medium and heavy nuclei formed in heavy ion fusions~\\cite{Schiller}, deep inelastic scattering of light particles on heavy targets~\\cite{Baumann,Heckman}, and $\\alpha$ induced fusions~\\cite{Kolkata}. The common features of the hot GDR are: (1) Its energy is nearly independent of $T$ and $J$, (2) Its full width at half maximum (FWHM) remains mostly unchanged in the region of $T\\leq$ 1 MeV, but increases sharply with $T$ within 1$\\leq T\\leq$ 2.5 - 3 MeV, and seems to saturate at $T\\geq$ 4 MeV. As a function of $J$, a significant increase in the GDR width is seen only at $J\\geq$ 25 - 27$\\hbar$. In Ref. \\cite{Kusnezov} some GDR data were reanalyzed by adding the pre-equlibrium $\\gamma$ emission  and it was claimed that the GDR width does not saturate. However, it was realized later that the pre-equilibrium emission is proportional to the asymmetry between projectiles and targets and lowers the CN excitation energy, which alters the conclusion on the role of pre-equlibrium emission. The recent measurements in $^{88}$Mo at $T\\geq 3$ MeV and $J>$ 40$\\hbar$ did not show any significant effect of pre-equilibrium emission on the GDR width~\\cite{Maj}.   The evaporation width owing to the quantal mechanical uncertainty in the energies of the CN states was also proposed to be added into the total GDR width~\\cite{Chomaz}. However, the high-energy $\\gamma$-ray spectra resulting from the complete {\\scriptsize CASCADE} calculations~\\cite{Gervais} including the evaporation width turned out to be essentially identical to those obtained by neglecting this width even up to excitation energy higher than 120 MeV for $^{120}$Sn (i.e., at $T >$  3.3 MeV). This indicates that the effect of evaporation width, if any, may become noticeable only at much higher values of $T$ ($\\gg$ 3.3 MeV) and $J$ ($\\gg$ 30$\\hbar$). \nIn a classical representation of the GDR as a damped spring mass system, the damping width of the oscillator (the GDR width) should be smaller than its frequency (the GDR energy) otherwise the spring mass system cannot make any oscillation. This means that the GDR width is upper-bounded by its energy. This implies the saturation of the GDR width.\n\nThe present lecture summarizes the achievements of the Phonon Damping Model (PDM)~\\cite{PDM1a,PDM2,PDMJ} in the description of the the GDR width and shape at finite $T$ and $J$ (Sec. \\ref{TJ}). As two applications, the GDR parameters predicted by the PDM and experimentally extracted are used to calculate the shear viscosity of finite hot nuclei, which is also employed to test the recent preliminary data of the GDR width at high $T$ and $J$ in $^{88}$Mo (Sec. \\ref{visco}). Conclusions are drawn in the last section.\n\n\\section{Damping of GDR in highly excited nuclei}\n\\label{TJ}\n\\subsection{GDR width and shape in hot nuclei}\n\\label{T}\nThe width of the GDR built of the ground state ($T=$ 0) (the quantal width $\\Gamma_{Q}$), consists of the three components: (i) the Landau width $\\Gamma^{LD}$, which is essentially the variance $\\sqrt{\\langle E^2\\rangle - \\langle E\\rangle^2}$ of the $ph$-state distribution, (ii) the spreading width $\\Gamma^{\\downarrow}$ caused by coupling of $1p1h$ states to more complicated configurations such as $2p2h$ ones, and (iii) the escape width $\\Gamma^{\\uparrow}$ owing to the direct particle decay into hole states of the residual nucleus because of coupling to continuum. In medium and heavy nuclei, the major contribution to $\\Gamma_Q$ is given by $\\Gamma^{\\downarrow}$, whereas $\\Gamma^{LD}$ and $\\Gamma^{\\uparrow}$ account for a small fraction. The calculations within the microscopic models such as the particle+vibration model~\\cite{Bortignon} and the quasiparticle-phonon model~\\cite{QPM} have shown that $\\Gamma^{\\downarrow}$ does not increase width $T$. Therefore the mechanism of the width's increase width $T$ should be sought beyond the one that causes $\\Gamma^{\\downarrow}$.\n\nThe PDM's Hamiltonian consists of the \nindependent single-particle (quasiparticle) field, GDR phonon field, and the coupling between \nthem [Eq. (1) in Ref. \\cite{PDM1a}].   The Woods-Saxon potentials at $T =$ 0 are often \nused to obtain the single-particle energies $\\epsilon_k$. The GDR width $\\Gamma(T)$ is a sum: $\\Gamma(T)=\\Gamma_{\\rm Q}+\\Gamma_{\\rm T}$ of\nthe quantal width, $\\Gamma_{\\rm Q}$, and thermal width, $\\Gamma_{\\rm T}$.\nIn the presence of superfluid pairing, the quantal and thermal widths \nare given as~\\cite{PDM2}\n\\begin{equation}\n\\Gamma_{\\rm Q}=2\\gamma_Q(E_{GDR})=2\\pi F_{1}^{2}\\sum_{ph}[u_{ph}^{(+)}]^{2}(1-n_{p}-n_{h})\n\\delta[E_{\\rm GDR}-E_{p}-E_{h}]~,\n\\label{GammaQ}\n\\end{equation}\n\\begin{equation}\n\\Gamma_{\\rm T}=2\\gamma_T(E_{GDR})=2\\pi F_{2}^{2}\\sum_{s>s'}[v_{ss'}^{(-)}]^{2}(n_{s'}-n_{s})\n\\delta[E_{\\rm GDR}-E_{s}+E_{s'}]~, \n\\label{GammaT}\n\\end{equation}\nwhere $u_{ph}^{(+)} = u_pv_h+u_hv_p$, $v_{ss'}^{(-)}=u_su_{s'}-v_sv_{s'}$ ($ss' = pp', hh'$) with\n$u_k$ and $v_k$ being the coefficients of Bogolyubov's transformation from particle operators to \nquasiparticle ones, $E_k\\equiv\\sqrt{(\\epsilon_k-\\lambda)^2+\\Delta^2}$, with superfluid pairing gap $\\Delta$,\nare quasiparticle energies, $n_k$ are quasiparticle occupations numbers, which, for medium and heavy nuclei, can be well approximated with the Fermi-Dirac distribution for\nindependent quasiparticles, $n_k = [\\exp(E_k\/T)+1]^{-1}$. The parameter $F_1$ is chosen so that $\\Gamma_Q$ at $T=$ 0 is equal to GDR's width at $T=$ 0, whereas the parameter $F_2$ is chosen so that, with varying $T$,  the GDR energy $E_{GDR}$ does not change significantly. The latter is found as the solution of the equation $E_{GDR} - \\omega_{q}-P_q(E_{GDR})=0$, where $\\omega_q$ is the energy of the GDR phonon before the coupling between the phonon and single-particle mean fields is switched on, and $P_q(\\omega)$ is the polarization operator owing to this coupling, whose explicit expression in given in Refs. \\cite{PDM2}. The GDR strength function is calculated as \n\\begin{equation}\nS_{q}(\\omega) = \\frac{1}{\\pi}\\frac{\\gamma_Q(\\omega) + \\gamma_T(\\omega)}{(\\omega-E_{GDR})^2+[\\gamma_Q(\\omega) + \\gamma_T(\\omega)]^{2}}~.\\label{S}\n\\end{equation}\nIn numerical calculations the representation $\\delta(x) =\\lim_{\\varepsilon\\rightarrow 0}\\varepsilon\/[\\pi(x^{2}+\\varepsilon^2)]$ is used for the $\\delta$-functions in Eqs. (\\ref{GammaQ}) and (\\ref{GammaT}) with $\\epsilon=$ 0.5 MeV.\n\n\\begin{figure}\n\\center{\n\\includegraphics[width=10.3cm]{widthSnPb.eps}}\n\\caption{GDR widths for $^{120}$Sn (a) and $^{208}$Pb (b) predicted by the PDM (thick solid), pTSFM (dot-dashed), AM (double dot-dashed), and FLDM (thin solid) as functions of $T$ in comparison with experimental data in tin and lead regions. (c): Exact canonical neutron (N) and proton (Z) pairing gaps for $^{201}$Tl as functions of $T$. (d): GDR width for $^{201}$Tl obtained within the PDM as a function of $T$ (thick solid) including the exact canonical gaps in (c) in comparison with the experimental data for $^{201}$Tl (black circles) and $^{208}$Pb (open boxes). The thin solid line is the PDM result without the effect of thermal pairing. The dotted line is the PDM result for $^{208}$Pb [the same as the thick solid line in (b)]. \n\\label{widthSnPb}}\n\\end{figure}\nThe GDR widths predicted by the PDM, the two versions of thermal shape fluctuation model (TSFM), namely the phenomenological TSFM (pTSFM)~\\cite{pTSFM} and the adiabatic model (AM)~\\cite{AM}, and the Fermi liquid drop model (FLDM)~\\cite{FLDM} for $^{120}$Sn and $^{208}$Pb are shown in Figs. \\ref{widthSnPb} (a) and \\ref{widthSnPb} (b) in comparison with the experimental systematics. The PDM results for $^{120}$Sn include the effect of non-vanishing thermal pairing gap because of thermal fluctuations owing to finiteness of nuclei. The figure clear shows that among the models under consideration, the PDM is the only one that is able to describe well the experimental data in the entire temperature region including $T\\leq$ 1 MeV, where the other model fail. It is also able to reproduce the very recent data for the GDR width in $^{201}$Tl at 0.8$\\leq T<$ 1.2 MeV [Fig. \\ref{widthSnPb} (d)] after including the exact canonical \ngaps for neutrons and protons shown in Fig. \\ref{widthSnPb} (c)~\\cite{Tl201}.\n\n\\begin{figure}\n\\center{\n\\includegraphics[width=9.3cm]{SE.eps}}\n\\caption{Experimental (shaded areas) and theoretical shapes for GDR in $^{120}$Sn generated by the {\\scriptsize CASCADE} code at various excitation energies. (a) -- (c): predictions generated by using the PDM strength functions. (d) and (e)~\\cite{Gervais}: dotted and dashed lines are generated by using the TSFM strength functions, exhausting 80$\\%$ and 100$\\%$ TRK sum rule, respectively.\\label{SE}}\n\\end{figure}\nFor an adequate description of not only the width but also the entire GDR shape, the PDM strength functions were incorporated into all the decay steps of the full statistical calculations and the generated results are compared with those obtained from the measured $\\gamma$-ray spectra in Fig. \\ref{SE}, which shows that the PDM describes fairly well the GDR shape [Fig. \\ref{SE} (a)], whereas the TSFM fails in doing so [Fig. \\ref{SE} (b)].\n\\subsection{GDR width and shape in hot and rotating nuclei}\n\\label{J}\nTo describe the non-collective rotation of a spherical nucleus, the $z$-projection $M$ of the total angular momentum $J$ is added into the PDM Hamiltonian as  $- \\gamma\\hat{M}$, where $\\gamma$ is the rotation frequency~\\cite{PDMJ}. The latter and the chemical potential are defined, in the absence of pairing,  from the equation $M = \\sum_k m_k(f_{k}^{+}-f_{k}^{-})~,$ and  $N =\\sum_k(f_{k}^{+}+f_{k}^{-})~$, where $N$ is the particle number and $f_k^{\\pm}$ are the single-particle occupation numbers, $f_{k}^{\\pm} =1\/[\\exp(\\beta E_k^{\\mp})+1]$, and $E_k^{\\mp} = \\epsilon_k-\\lambda\\mp\\gamma m_k~$. With the smoothing of $\\delta$-functions by using the Breit-Wigner distribution mentioned in Sec. \\ref{T}, the final form of phonon damping $\\gamma_q(\\omega)$ becomes\n\\begin{equation}\n\\gamma_q(\\omega) = \\varepsilon\\sum_{kk'}[{\\cal F}_{kk'}^{(q)}]^{2}\n\\bigg[\\frac{f_{k'}^{+}-f_{k}^{+}}\n{(\\omega-E_k^{-} + E_{k'}^{-})^2+\\varepsilon^2}+\\frac{f_{k'}^{-}-f_{k}^{-}}{(\\omega-E_k^{+} + E_{k'}^{+})^{2}+\\varepsilon^{2}}\\bigg]~,\n\\label{gamma1}\n\\end{equation}\nwhere $(k,k') = ph, pp', hh'$. The GDR strength function is calculated by using the same Eq. (\\ref{S}) where \n$\\gamma_Q(\\omega) +\\gamma_T(\\omega)$ is replaced with $\\gamma_q(\\omega)$. The explicit expression for the polarization operator $P_q(\\omega)$ is given in Eq. (13) of Ref. \\cite{PDMJ}.\n\n\\begin{figure}\n     \\center{\\includegraphics[width=9cm]{streJSn1.eps}\n     \\caption{(a) -- (f): GDR strength functions  for\n     $^{106}$Sn at $T=$ 0.5, 1, 2, 3, 4, and 5 MeV as shown at the curves and $M=$ 0, 10, 20, 40, 60, and 80$\\hbar$ as shown in the panels.\n     (g): FWHM of GDR for $^{106}$Sn as a function of $T$\n     at several values of $M$ (in $\\hbar$) shown at the curves, (h): FWHM of GDR for $^{106}$Sn as a function of $M$ at several values of $T$ (in MeV) shown at the curves. The experimental data for GDR in $^{106}$Sn (solid circles), $^{109,110}$Sn (solid and open boxes) are adapted from Refs. \\cite{Mattiuzzi,Bra}.}    \n      \\label{streJSn}}\n\\end{figure}\nShown in  Fig. \\ref{streJSn}  are the GDR strength functions $S(\\omega)$ and the widths in\n$^{106}$Sn at various $T$ and $M$.  The GDR shape becomes smoother  as $T$ and $M$ increase, and the smoothing caused by the angular momentum is stronger than that caused by thermal effects (Figs. \\ref{streJSn} (a) -- \\ref{streJSn} (f)). The GDR width increases with both $T$ and $M$ and stronger at low $T$ and $M$. This increase in the width approaches a saturation at moderate and high $T$ and\/or $M$. As a function of $T$, the saturation begins at $T>$ 4 MeV in $^{106}$Sn [Fig. \\ref{streJSn} (g)], whereas as a function of $M$ it takes place in $^{106}$Sn already at $T\\geq$ 3 MeV [Fig. \\ref{streJSn} (h)].  Experimental data for $^{106}$Sn~\\cite{Mattiuzzi} and $^{109,110}$Sn~\\cite{Bra} are also shown in Figs. \\ref{streJSn} (g) and \\ref{streJSn} (h), which are in fair agreement with theory.\n\\section{Shear viscosity of hot nuclei}\n\\label{visco}\nIn the verification of the condition for applying hydrodynamics to \nnuclear system, the quantum mechanical \nuncertainty principle requires a finite viscosity for any thermal \nfluid. Kovtun, Son and Starinets (KSS)~\\cite{KSS} conjectured\nthat the ratio $\\eta\/s$ of shear viscosity $\\eta$ to the entropy \nvolume density $s$ is bounded below for all fluids, namely the value\n${\\eta}\/{s}= {\\hbar}\/(4\\pi k_{B})$  \nis the universal lower bound (KSS bound or unit).\nFrom the viewpoint of collective theories, one of the fundamental \nexplanations for the giant resonance damping is the friction term (or viscosity) \nof the neutron and proton fluids.  By using the Green-Kubo's relation, it has been shown in Ref. \\cite{visco} that the shear viscosity $\\eta(T)$ \nat finite $T$ is expressed in terms of the GDR's parameters at zero and finite $T$ as\n\\begin{equation}\n\\eta(T)=\\eta(0)\\frac{\\Gamma(T)}{\\Gamma(0)}\n\\frac{E_{GDR}(0)^{2}+[\\Gamma(0)\/2]^{2}}{E_{GDR}(T)^{2}+[\\Gamma(T)\/2]^{2}}~.\n\\label{eta1}\n\\end{equation}\n\\begin{figure}\n     \\center{\\includegraphics[width=9.0cm]{etaratio.eps}\n     \\caption{Shear viscosity $\\eta(T)$ [(a) and (b)] and ratio \n     $\\eta\/s$ [(c) and (d)] as functions of $T$ for nuclei in tin [(a) and (c)], and lead [(b) and (d)] regions. The  \n     gray areas are the PDM predictions by using $0.6u\\leq\\eta(0)\\leq \n     1.2u$ with $u=$ 10$^{-23}$ Mev s fm$^{-3}$.}\n     \\label{eta&ratio}}\n\\end{figure}\nThe predictions for the shear viscosity $\\eta$ and the ratio $\\eta\/s$ by the PDM,  pTSFM, AM, and \n FLDM for $^{120}$Sn and $^{208}$Pb are plotted as functions of $T$ \nin Fig. \\ref{eta&ratio} in comparison with the empirical results. \nThe latter are extracted from the \nexperimental systematics for GDR in tin and lead regions~\n\\cite{Schiller} making use of Eq. (\\ref{eta1}).  It is seen in Fig. \\ref{eta&ratio} that the predictions by the PDM have the best overall \nagreement with the empirical results. It produces an increase \nof $\\eta(T)$ with $T$ up to 3 - 3.5 MeV and a saturation of $\\eta(T)$ within (2 - 3)$u$ \nat higher $T$ [with \n$\\eta(0)=$ 1$u$, $u=$ 10$^{-23}$ Mev s fm$^{-3}$]. The ratio $\\eta\/s$ decreases sharply with \nincreasing $T$ up to $T\\sim$ 1.5 MeV, starting from which the decrease \ngradually slows down to reach (2 - 3) KSS units \nat $T=$ 5 MeV. The FLDM has a similar trend as that of the PDM up to \n$T\\sim$ 2 - 3 MeV, but at higher $T$ ($T>$ 3 MeV for $^{120}$Sn or 2 MeV for \n$^{208}$Pb) it produces an increase of both $\\eta$ and $\\eta\/s$ with $T$. \nAt $T=$ 5 MeV the FLDM model predicts the ratio $\\eta\/s$ within (3.7 - 6.5) KSS units, which are \nroughly 1.5 -- 2 times larger than the PDM predictions. The AM and pTSFM show a similar trend for $\\eta$ and $\\eta\/s$. \nHowever, in order to obtain such similarity, $\\eta(0)$ in the pTSFM \ncalculations has to be reduced to 0.72$u$ instead of 1$u$. They all \noverestimate $\\eta$ at $T<$ 1.5 MeV. \n\nA model-independent estimation for the high-$T$ limit of the ratio $\\eta\/s$ can also be inferred \ndirectly from Eqs. (\\ref{eta1}). Assuming that,\nat the highest $T_{max}\\simeq$ 5 - 6 MeV where the GDR can still exist, the \nGDR width $\\Gamma(T)$ cannot exceed $\\Gamma_{max}\\simeq \n3\\Gamma(0)\\simeq 0.9E_{GDR}(0)$~\\cite{Auerbach1}, and $E_{GDR}(T)\\simeq E_{GDR}(0)$, one obtains from \nEq. (\\ref{eta1}) $\\eta_{max} \\simeq 2.551\\times\\eta(0)$. By noticing \nthat, $S_{F}\\to 2\\Omega\\ln{2}$ at $T\\to\\infty$ \nbecause $n_{j}\\to$ 1\/2, where $\\Omega=\\sum_{j}(j+1\/2)$ for \nthe spherical single-particle basis or sum of all doubly-degenerate levels for the \ndeformed basis and that the particle-number conservation requires $A = \n\\Omega$ since all single-particle occupation numbers are equal to \n1\/2, one obtains the high-$T$ limit of entropy density $s_{max} = 2\\rho\\ln{2}\\simeq 0.222~(k_{B})$.\nDividing $\\eta_{max}$ by $s_{max}$ yields the high-$T$ limit (or lowest bound) for $\\eta\/s$ \nin finite nuclei, that is $({\\eta}\/{s})_{min}\\simeq 2.2^{+0.4}_{-0.9}$ KSS units, where the empirical values for $\\eta(0) = \n1.0^{+0.2}_{-0.4}~u$ are used~\\cite{Auerbach1,fission}. Based on these results, one can conclude that\nthe value of $\\eta\/s$ for medium and heavy nuclei at $T=$ 5 MeV is in \nbetween (1.3 - 4.0) KSS units, which is about (3 - 5) times smaller \n(and of much less uncertainty) that the value between (4 - 19) KSS units predicted by \nthe FLDM for heavy nuclei~\\cite{Auerbach}, where the same lower value $\\eta(0)=$0.6$u$ was used.\n\nFinally, by using the temperature dependence of $\\eta\/s$ and the KSS lower bound conjecture, it is possible to examine the recent preliminary data for the GDR width in $^{88}$Mo in Ref. \\cite{Maj}. Shown in Fig. \\ref{Stest} is the strength function \n$S_L(\\omega) = {\\omega}[S(\\omega, E_{GDR})-S(\\omega, -E_{GDR})]\/E_{GDR}$, where $S(\\omega, \\pm E_{GDR})$ are the PDM strength functions (\\ref{S}) at finite $T$ and $J$ for the GDR located at $\\pm E_{GDR}$. The PDM predictions are shown at the initial temperature $T_{max}$ of the compound nucleus ($T_{max} =$ 3 and 4 MeV in Figs. \\ref{Stest} (a) and \\ref{Stest} (b), respectively), and also at $T=$ 2.5  MeV (Figs. \\ref{Stest} (a)) and 3.2 MeV (Figs. \\ref{Stest} (b)), i.e. within the error bars of the average temperature $\\langle T\\rangle$ obtained by averaging over all the GDR decay steps ($\\langle T\\rangle =$ 2$\\pm$0.6 and 2.6$\\pm$0.8 MeV for $E^{*}=$ 300 and 450 MeV, respectively~\\cite{Maj1}). While the PDM strength functions and experimental line shapes of the GDR agree fairly well at $M=$ 41 $\\hbar$ with the FWHM $\\Gamma$ predicted by the PDM between 9.6 MeV (T = 2.5 MeV) and 11 MeV (T = 3 MeV), they strongly mismatch at $M=$ 44 $\\hbar$, where the experimental GDR peak becomes noticeably narrower with a width $\\Gamma_{ex}\\simeq$ 7.5 MeV. By using this value $\\Gamma_{ex}$ and $\\eta(0)=$ 0.6 $u$, one ends up with the value of $\\eta\/s=$ 0.85 KSS units. Including the error bars in $\\Gamma_{ex}$ leads to $\\Gamma_{ex}^{<}\\simeq$ 6 MeV and $\\Gamma_{ex}^{>}\\simeq$ 8.5 MeV, which give the values of $\\eta\/s$ equal to 0.69 and 0.94 KSS, respectively. All these values are smaller than the KSS lower bound conjecture. This may indicate that either (i) the data analysis in extracting the experimental GDR strength function for $^{88}$Mo at $E^{*} = 450$ MeV (Fig. \\ref{Stest} (b)) is inaccurate, or (ii) a violation of the KSS conjecture has been experimentally confirmed for the first time ever. The reanalysis of the data is now underway to clarify which one from these two conclusions holds~\\cite{Maj1}. \\begin{figure}\n\\center{\n     \\includegraphics[width=11 cm]{STE1_rev.eps}\n     \\caption{GDR strength function $S_L(\\omega)$ for $^{88}$Mo at $M=$ 41 $\\hbar$ (a) and \n    $M=$ 44 $\\hbar$ (b) predicted by the PDM in comparison with the preliminary data from Ref. \\cite{Maj}.}     \n      \\label{Stest}}\n\\end{figure}\n \n\n\\section{Conclusions}\nThe PDM generates the damping of GDR through its couplings to $ph$ configurations, causing the quantal width, as well as to $pp$ and\/or $hh$ configurations, causing the thermal width. This leads to an overall increase in the GDR width at low and moderate $T$, and its saturation at high $T$. At very low T $<$ 1 MeV the GDR width remains nearly constant because of thermal pairing. The GDR width also increases with angular momentum $M$ and saturates at high $M$, but this saturation goes beyond the value of maximal angular momentum that the nucleus can sustain without violating the KSS conjecture, that is 46 and 55$\\hbar$ for $^{88}$Mo and $^{106}$Sn, respectively, if the value $\\eta(0) =$ 0.6$\\times 10^{-23}$ Mev s fm$^{-3}$ for the shear viscosity at $T=$ 0 is used.   The PDM predictions agree well with the experimental systematics for the GDR width and shape in various medium and heavy nuclei. The PDM also predicts the shear viscosity to the entropy-density ratio $\\eta\/s$ between (1.3 - 4.0) KSS units for medium and heavy nuclei at $T=$ 5 MeV, almost the same at that of the quark-gluon-plasma like matter at $T>$ 170 MeV discovered at RHIC and LHC. The PDM and the KSS conjecture are also used to show that the recent preliminary experimental data for GDR in $^{88}$Mo~\\cite{Maj} need to be reanalyzed. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{secIntro}\n\nThe Peak Over Threshold (POT) method, described in details by \\citet[][Chapter 4]{coles2001introduction},  \nhas been used in many fields to identify extremal events such as loads,\nwave heights, floods, wind velocities, etc. This method provides a\nmodel for independent exceedances over a high threshold.\nAccording the  \nPickands-Balkema-de Haan theorem \\citep{balkema:dehaan:1974,pickands:1975}\nfor a threshold $u$  sufficiently high, the unnormalized conditional tail distribution of a random variable $Y$ can be approximated as \n$$\n\\Pr(Y-u\\le y|Y>u)\\approx \tH_{\\xi}(y\/\\psi_u),\\qquad \\text{for}\\, y\\ge 0\\quad \\text{and}\\quad \\psi_u >0\n$$\nwhere\n\\begin{equation}\\label{eq:GPD}\n\tH_{\\xi}(z) =\n\t\\begin{cases}\n\t\t1 - {(1+\\xi y)}_+^{-1\/\\xi} & \\text{for $\\xi \\neq 0$}\\\\\n\t\t1-\\exp\\{-y \\}& \\text{for $\\xi = 0$}\n\t\\end{cases}  \n\\end{equation}\nand  $a_+ = \\max (a, 0)$. \n\n The limiting distribution \\eqref{eq:GPD} is referred to as the Generalized Pareto  distribution (GPD) and the parameters $\\xi$ and $\\psi_u$ are respectively the shape and the scale parameters. \n The scale parameter depend on the threshold $u$, but \nin  remainder of the paper the subscript $u$ is  dropped in the  notation.\n\nModeling using GPD has become very popular since the paper of \\citet{davison:smith:1990} where estimation and model-checking procedures for univariate and regression data are developed.\n Such procedure have then been extended  by allowing the GPD parameters to be represented as smooth functions of covariates \\citep{chavez-demoulin:davison:2005}.\n \n Since then several R packages have been developed to fit GPDs  and contributed to the Comprehensive R Archive Network (CRAN) in the field of extreme value analysis \\citep{team2013r}. Among them, \\texttt{ismev} \\citep{heffernan2016ismev}, providing tools for the  extreme value analyses presented in \\citet{coles2001introduction}, \\texttt{evd} \\citep{stephenson2002evd}, \\texttt{evir} \\citep{pfaff2018package}, \\texttt{extRemes} \\citep{gilleland2016extremes} and \\texttt{mev} \\citep{belzile1mev}, have offered various approaches for fitting univariate and multivariate GPDs. \\citet{gilleland2013software} and, more recently, \\citet{Belzile-et-al:2022} review them while the \\citet{dutang2022cran}'s CRAN Task View provides an up-to-date list of the R packages in this field. In particular, some of them offer regression-based models for extreme distributions, where GPD parameters are allowed to vary with covariates. \nBeyond the linear forms introduced in the packages \\texttt{ismev} and \\texttt{evd},  \na few R packages allow to fit GPDs with parameters following additive forms i.e. when the parameters or their transformations can be modelled as additive smooth functions of the covariates. \nIn the sequel we term this form a Generalized Additive Model (GAM) \\citep{Hastie:Tibshirani:1990,wood2017generalized}. Thus \\texttt{VGAM} \\citep{yee2007vector}, \\texttt{gamlss} \\citep{rigby2005generalized} and \\texttt{evgam} \\citep{youngman2020evgam} provide this functionality for GPDs, as reviewed among other alternatives in \\citet{youngman2020evgam}. \n\n\nIt is clear that GPD  needs\nthe determination of a threshold which is neither too high.\nThe choice of an appropriate threshold is  challenging because the stability plots and the mean residual life\n\\citep[][Chapter 4]{coles2001introduction} plot do not give an obvious value of a suitable threshold. \nThe problem is tackled  in a regression setting by\n\\citet{eastoe:tawn:2009}. \nHowever the question of the threshold choice and then simulation of the whole spectrum of the random variable considered remains problematic. \n\nForming mixtures of Exponential distributions \\citep{wilks1998multisite,keller2015implementation} or conditional Gamma distributions \\citep{kenabatho2012stochastic,chen2014multi} remains a competitive alternative e.g. for heavy-tailed precipitation modeling and simulation. Approaches of this type are implemented in the package \\texttt{GSM}, that uses a Bayesian approach for estimating a mixture of Gamma distributions in which the mixing occurs over the shape parameter and the procedure only requires to specify a prior distribution for a single parameter. The package \\texttt{evmix} fits an extreme value mixture model with normal behavior for the bulk distribution up to the threshold and conditional GPD above the threshold. Options for profile likelihood estimation for threshold and fixed threshold approach are provided. Finally, \\texttt{distributionsrd} provides the double Pareto-lognormal distribution. \n\n\nAlthough interesting, in practice these packages do not allow to fit nonlinear models as flexible as the GAM options allowed for the GPD alone.\nBesides, the mixture approaches tend to quickly inflate the number of parameters and the inference of the latter may be problematical operationally \\citep{naveau2016modeling}.\n\n\\citet{papastathopoulos2013extended} opened another route in the field of heavy-tail distribution modeling by replacing the uniform draws in the inverse cumulative distribution function (CDF) transform to generate simulations of the GPD, by draws from a richer distribution e.g. type Beta distribution. Using this approach, \\citet{naveau2016modeling} presented a concise parametric formulation (later extended to a semi-parametric formulation \\citep{tencaliec2020flexible}) for simulating the whole range of a positive random variable (namely precipitations), that makes use of GPD to model not only the high quantiles but the smallest ones as well and bypass the issue of threshold selection. \nThe main idea rooted from composing a cumulative distribution function with the GPD (see section 2 for more details). Very recently, a similar idea was taken up by \\citet{stein2021env,stein2021ext}.\n\n\nIn the sequel, we refer to the extension of GPD in the sense of \\citet{naveau2016modeling} as Extended Generalized Pareto distribution (EGPD).\nAn implementation of EGPD with four parametric families, together with the fitting procedure, can be found  in the R package \\texttt{mev}.  \n\n\nThe aim of this work is to extend the applicability of the EGPD in the presence of covariates that can modify the parameters of the distribution.\nThe extension takes place in the same spirit as  GAM. \nWe also want to present an add-on to the \\texttt{gamlss} package that allows to make inference on the parameter.\n\nWe have chosen the \\texttt{gamlss} package because it provides univariate distributional regression models, where parameters of the assumed distribution for the response can be modeled as additive functions of the covariates. It is flexible enough to allow the encoding of novel families of distribution, as long as their density, cumulative, quantile and random generative function and, optionally, first and second (cross-) derivatives of the likelihood can be computed.  We seize this potential to implement the parametric EGPD family in a generic way, allowing to introduce new parametric forms, up to 4 parameters so far (due to limitations of \\texttt{gamlss}), satisfying the requirement of the EGPD family.\n\nThe remaining of this article will first introduce the EGPD and the  distribution  regression model (Section \\ref{secTheory}), followed by the key commands to use the EGPD within the \\texttt{gamlss} package and our add-on in Section \\ref{secTuto}. \nSection \\ref{secApplRainfalls} implements them to perform an analysis of hourly positive rainfalls on the North-west of France and discuss specific advantages and limitations of distributional regression modeling and its implementation in R, while Section \\ref{secConclu} concludes the paper. \n\n\\section{A distributional regression model}\n\\label{secTheory}\n\n\\subsection{Extended-GPD}\n\\label{secEGPDmod}\n\nThe EGPD \\citep{naveau2016modeling}  accounts simultaneously and in a parsimonious way for both extreme and non extreme values of a random variable $Y$. Its cumulative distribution function (CDF) reads:\n\\begin{equation}\n\\label{CDFegpd}\n    F(y) = G\\{ H_{\\xi}(y\/\\psi) \\} \\:, \\: \\: \\text{for all}\\quad y > 0.\n\\end{equation}\nThe function $G(\\cdot)$ is a continuous CDF on the unit interval. It is constrained by several assumptions: 1) we need to ensure that the upper-tail behavior of $F$ follows a GPD; 2) the low values of $Y$, seen as the upper tail of $-Y$ when $y \\rightarrow 0$, follows a GPD as well. \n\nThis leads to the following consequences for any candidate CDF $G(\\cdot)$: \n\\begin{enumerate}\n\t\\item[a.]  the upper-tail behavior\n\tof $1-F(y)$ is equivalent to the original GPD tail used to build $F(y)$ ; \n\t\\item[b.]  when $y \\rightarrow 0$, $\\displaystyle F(y) \\sim \\frac{c}{\\psi^s}y^s $ where $\\displaystyle c = \\lim_{y \\rightarrow 0} \\frac{G(y)}{y^s}$ is positive and finite for some real $s$.\n\\end{enumerate}\n\nFour parametric families, $G(\\cdot,\\kappa)$, $\\kappa=(\\kappa_1,\\ldots,\\kappa_k)^T$, that satisfies the above-mentioned constraints are proposed in \\citet{naveau2016modeling}.\n\n\\begin{enumerate}\n    \\item \\label{c1} \n    $G(u;\\kappa)=u^{\\kappa_1}$, $\\kappa_1 >0$\n    \\item \\label{c2}\n    $G(u;\\kappa)=\\kappa_3 u^{\\kappa_1}+(1-\\kappa_3)u^{\\kappa_2}$, $\\kappa_2 \\geq \\kappa_1 >0$ and $\\kappa_3 \\in [0,1]$\n    \\item \\label{c3}\n    $G(u;\\kappa)=1-Q_{\\kappa_1}\\{(1-u)^{\\kappa_1}\\}$, $\\kappa_1>0$ where $Q_{\\kappa_1}$ is the CDF of a Beta random variable with parameters $1\/\\kappa_1$ and $2$, that is:\n    $$Q_{\\kappa_1}(u)=\\frac{1+\\kappa_1}{\\kappa_1} u^{1\/\\kappa_1} \\Big( 1-\\frac{u}{1+\\kappa_1} \\Big)$$\n    \\item \\label{c4}\n    $G(u;\\kappa)=[1-Q_{\\kappa_1}\\{(1-u)^{\\kappa_1}\\}]^{\\kappa_2\/2}$, $\\kappa_1, \\kappa_2>0$\n\\end{enumerate}\n\nIn model \\ref{c1}, $\\xi$ controls the rate of upper tail decay, $\\kappa_1$ the shape of the lower tail and $\\psi$ is a scale parameter. $\\kappa_1=1$ allows us to recover the GPD model, while varying it gives flexibility in the description of the lower tail. \n\nModel \\ref{c2}, as a mixture of power laws, allows to increase the flexibility provided by model \\ref{c1}. Therein, $\\kappa_1$ controls the lower tail behavior while $\\kappa_2$ modifies the shape of the density in its central part. \n\nModel \\ref{c3} is connected to the work of \\citet{falk2010laws}. $\\xi$ keeps controlling the upper extreme tail, $\\kappa_1$ the central part of the distribution in a threshold tuning way. However, this model imposes a behavior type $\\chi^2$ for the lower tail (null density at $0$) instead of allowing the data to constrain it. \n\nHence Model \\ref{c4}, which adds an extra parameter $\\kappa_2$ to circumvent this limitation. In this last model, $\\kappa_2, \\kappa_1, \\xi$ control respectively the lower, moderate and upper parts of the distribution. By construction, the lower and upper tails are GPD of shape parameters $\\kappa_2$ and $\\xi$ respectively. \n\n\nIn the sequel we denote the vector of parameters $(\\xi,\\psi,\\kappa^T)^T$ with $\\theta = (\\theta_1, ..., \\theta_d)^T$ and the density of a parametrized EGPD with $EGPD(y;\\theta)$.\nTo simulate from Equation (\\ref{CDFegpd}), the user randomly draws a uniform variable $U$ in $[0,1]$ and then applies the quantile function $Y=F^{-1}(U)$, given for $p \\in [0,1]$ by:\n\\begin{equation}\n\\label{quantileegpd}\n     y_p=F^{-1}(p) =\n     \\begin{cases}\n       \\frac{\\psi}{\\xi}[\\{ 1-G^{-1}(p) \\}^{-\\xi} - 1] & \\text{for $\\xi \\neq 0$}\\\\\n       -\\psi \\log{ \\{ 1-G^{-1}(p) \\}} & \\text{for $\\xi = 0$}\n    \\end{cases}  \n\\end{equation}\n\n\n\n\\subsection{Additive model specification}\n\\label{GAM}\n\nIn practice, the parameters of the distribution of $Y$ may depend on  some covariates $\\boldsymbol{x}=(x_1,\\ldots,x_m)^T$, i.e.\n\\begin{equation}\\label{eq:EGPD}\nY|\\boldsymbol{x}\\sim EGPD(\\cdot,\\theta(\\boldsymbol{x}))\n\\end{equation}\nwhere $\\theta(\\boldsymbol{x})=(\\theta_1(\\boldsymbol{x}),\\ldots,\\theta_d(\\boldsymbol{x}))^T$.\nThe previous specification is an instance  of a  distributional regression model \\citep{Stasinopoulos-et-al-2018}. \n\nFor relating the different distributional parameters  $(\\theta_1(\\boldsymbol{x}),\\ldots,\\theta_d(\\boldsymbol{x}))$ to the covariates, we rely on additive predictors of the form\n\\begin{equation}\\label{eq:predictors}\n\\eta_i(\\boldsymbol{x})=f_{i1}(\\boldsymbol{x})+\\cdots+f_{iJ_i}(\\boldsymbol{x})\t\n\\end{equation}\nwhere $f_{i1}(\\cdot),\\ldots, f_{iJ_i}(\\cdot)$ are smooth functions of the covariates $\\boldsymbol{x}$.\n\nThe representation \\eqref{eq:predictors} is particularly flexible and allows to capture complex dependence patterns between distribution parameters and covariates\n$\\boldsymbol{x}$. For instance a semiparametric model with  two covariates $(x_1,x_2)$ can be written as\n$$\n\\eta_i(\\boldsymbol{x})=\\beta_{i1}+ \\beta_{i2}x_1+f_{i2}(x_2)\n$$\nAnother example is when we want to model  space-time data observed at site $(s_1,s_2)$ and at time $t$. In this case $\\boldsymbol{x}=(s_1,s_2,t)^T$\n$$\n\\eta_i(\\boldsymbol{x})=f_{i1}(t)+ f_{i2}(s_1,s_2)\n$$\n\n\n\nIn analogy to Generalized Linear Model the predictors are  linked to the distributional parameters via known monotonic and twice differentiable link functions $h_i(\\cdot)$.\n\nIn the case of model 1, i.e. $G(u,\\kappa)=u^\\kappa_1$, common link functions are\n$$\n\\xi(\\boldsymbol{x})=\\eta_\\xi(\\boldsymbol{x}) \\mbox{ (i.e. the identity function)},\\quad \\psi(\\boldsymbol{x})=\\exp(\\eta_\\psi(\\boldsymbol{x})),\\quad \\kappa_1(\\boldsymbol{x})=\\exp(\\eta_{\\kappa_1}(\\boldsymbol{x})).\n$$\n\n\\begin{equation}\\label{eq:link}\n\\theta_i(\\boldsymbol{x})=h_i(\\eta_i(\\boldsymbol{x})), \\qquad i=1,\\ldots, d\n\\end{equation}\nThe functions $f_{ij}$in  \\eqref{eq:predictors} are approximated in terms of  basis function expansions\n\n\\begin{equation}\\label{eq:basis}\n\tf_{ij}(\\boldsymbol{x})=\\sum_{k=1}^{K_{ij}} \\beta_{ij,k}B_k(\\boldsymbol{x}),\n\\end{equation}\nwhere $B_k(\\boldsymbol{x})$ are the basis functions and $\\beta_{ij,k}$ denote the corresponding basis coefficients. These basis can be of different types, thoroughly presented e.g. in \\citet{stasinopoulos2017flexible} or \\citet{wood2017generalized}.\n\nThe basis function expansions can be written as \n$$\nf_{ij}(\\boldsymbol{x})=  \\boldsymbol{z}_{ij}(\\boldsymbol{x})^T\\boldsymbol{\\beta}_{ij}\n$$\nwhere $\\boldsymbol{z}_{ij}(\\boldsymbol{x})$ is still a vector of transformed covariates that depends on the basis functions.\nand $\\boldsymbol{\\beta}_{ij}=(\\beta_{ij,1},\\ldots, \\beta_{ij,K_{ij}})^T$ is a parameter vector to be estimated.\n\nTo ensure regularization of the functions $f_{ij}(\\boldsymbol{x})$  so-called penalty terms are added to the objective log-likelihood function. \nUsually, the penalty for each function $f_{ij}(\\boldsymbol{x})$ are quadratic penalty $\\lambda \\boldsymbol{\\beta}_{ij}^T \\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij}) \\boldsymbol{\\beta}_{ij} $ where $\\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij})$ is a known semi-definite matrix and the vector  $\\boldsymbol{\\lambda}_{ij}$ regulates the amount of smoothing\nneeded for the fit. A special case is when $\\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij})=\\lambda_{ij}\\boldsymbol{G}_{ij}$. Therefore the type and properties of the smoothing functions are controlled by the vectors  $\\boldsymbol{z}_{ij}(\\boldsymbol{x})$ and the matrices $\\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij})$.\n\nThe penalized log-likelihood function for the latter models reads:\n\\begin{equation}\\label{eq:penlik}\n\tl_p = l - \\frac{1}{2}\\sum_{i=1}^d \\sum_{j=1}^{J_i} \\boldsymbol{\\beta}_{ij}^T \\boldsymbol{G}_{ij}(\\boldsymbol{\\lambda}_{ij}) \\boldsymbol{\\beta}_{ij}\n\\end{equation}\nwhere $l$ represents the log-likelihood function for the model \\eqref{eq:EGPD}.\n\n\\section{Distributional regression  model for EGPD within the R package \\texttt{gamlss}}\n\\label{secTuto}\n\nIn this section we introduce an add-on script in R \nthat implements the EGPD regression model described in the previous section. \nThe script, available in\n\\texttt{github.com\/noemielc\/egpd4gamlss}, is flexible enough as it only requires the specification of a parametric CDF $G(\\cdot)$.\n\n\nThe Generalized Additive Model for Location, Scale and Shape (GAMLSS) is an example of distributional regression model and the companion package in R \\texttt{gamlss} \\citep{stasinopoulos2017flexible}  allows a straightforward fitting of multiple built-in and well-known family distributions with parameters that vary flexibly with $\\boldsymbol{x}$ as in \\eqref{eq:predictors}-\\eqref{eq:link}.\n\nAn interesting feature is the possibility to extend the family of distributions described in \\citet{rigby2019distributions}.\nThe only limitation is the dimension of the parameter vector $\\theta$, $d=4$,\nwhere the four parameters are denoted with $\\mu$, $\\sigma$, $\\nu$ and $\\tau$ in \\texttt{gamlss}.\n\nDue to the restriction on the number of parameters, our code allows the implementation of parametric distributions for $G$ with only 2 parameters, namely $\\kappa=(\\kappa_1,\\kappa_2)$.\nIn the following, the parameters $(\\xi,\\psi,\\kappa_1,\\kappa_2)$ of the EGPD distribution are identified with $(\\mu,\\sigma,\\nu,\\tau)$.\n\nThe generic implementation of a EGPD is provided in the file \\texttt{GenericEGPDg.R}. The \\texttt{MakeEGPD} function assembles a new EGPD family   distribution exploiting an user-built $G(\\cdot)$ as a function of $u$ and (at most) two parameters $\\nu$ and $\\tau$. \nThe required  first,  second and  cross derivatives of the density in \\eqref{eq:EGPD}, necessary for the fitting procedure in \\texttt{gamlss}, are obtained by   the symbolic derivation  using the R package \\texttt{Deriv}. Moreover the inverse function $G^{-1}(u)$ is  computed numerically. \n\n\nIn the following, we exemplify the use of \\texttt{MakeEGPD} by assembling   Model 1, $G(u;\\kappa)=u^{\\kappa_1}, \\kappa_1 > 0$.\n\n\\begin{lstlisting}[language=R]\nlibrary(gamlss)\nsource (\"GenericEGPDg.R\")\nEGPD1Family <- MakeEGPD (function (z,nu) z^nu, Gname = \"Model1\")\nEGPDModel1 <- EGPD1Family()\n\\end{lstlisting}\nThe last command is necessary and allows  to create the object EGPDModel1 (name concatenating \\texttt{EGPD} and \\texttt{Gname}) from the \\texttt{EGPD1Family} function.\n\nFor given values $\\mu_0, \\sigma_0$ and $\\nu_0$, the R functions of\ndensity, cumulative distribution function, quantile function and random generation are ready to use.\n\\begin{lstlisting}[language=R]\ndEGPDModel1(x,mu=mu0,sigma=sigma0,nu=nu0)\npEGPDModel1(x,mu=mu0,sigma=sigma0,nu=nu0)\nqEGPDModel1(u,mu=mu0,sigma=sigma0,nu=nu0)\nrEGPDModel1(n,mu=mu0,sigma=sigma0,nu=nu0)\n\\end{lstlisting}\n\nThe reserved names  for the distribution parameters in \\texttt{gamlss} are $\\mu$ \\texttt{mu} ($\\mu$), \\texttt{sigma} ($\\sigma$), \\texttt{nu} ($\\nu$), and \\texttt{tau} ($\\tau$).\nTo change the link functions $\\eta_i=g_i(\\theta_i)$ by means of which each EGPD parameter $\\theta_i$ (here $\\mu,\\sigma$ or $\\nu$) is indirectly optimised, or the default starting values for the parameters to fit, or to print these links, we use the commands:\n\n\\begin{lstlisting}[language=R]\nEGPD1Family(mu.link=\"identity\",mu.init=1, # log , inverse, own , ...\n            sigma.link=\"log\",sigma.init=3, nu.link=\"log\",nu.init=0.5)\nshow.link(EGPD1Family)\n\\end{lstlisting}\n\n\nIt is possible to create an original link function \"own\", by gathering the expressions of the link function $\\eta_i$, its inverse, the derivative of the latter w.r.t. parameter $\\theta_i$ and its area of validity. For instance to create a log-link function shifted to the right for $\\mu$:\n\n\\begin{lstlisting}[language=R]\n# the link function defining the predictor eta  as a function of the current distribution parameter (referred to as mu), i.e. eta = g(mu)\nown.linkfun <- function (mu) {\n    .shift = 0.0001\n    log (mu - .shift) \n}\n#  the inverse of the link function as a function of the predictor eta, i.e.mu = g-1 (eta)\nown.linkinv <- function (eta) {\n    shift = 0.0001 \n    thresh <- - log (.Machine$double.eps)\n    eta <- pmin (thresh, pmax(eta, -thresh))\n    exp (eta) + shift \n} \n# the first derivative of the inverse link with respect to eta\nown.mu.eta <- function (eta) {\n    shift = 0.0001  \n    pmax (exp (eta), .Machine$double.eps) \n}\n# the range in which values of eta are defined.\nown.valideta <- function (eta) TRUE\nEGPD1Family(mu.link=\"own\")\n\\end{lstlisting}\n\nFor more details on link function definition, we refer to   \\citet[][pag. 179]{stasinopoulos2017flexible}.\n\n\n\\section{Application to a hourly rainfall dataset in France}\n\\label{secApplRainfalls}\n\n\\subsection{Data}\n\nWe illustrate our add-on to package \\texttt{gamlss} with the analysis of the hourly precipitations recorded over the north-west area of France from 2016 to 2018 both included. This database, containing precipitations as well as other meteorological variables whose values are registered every 6 minutes over the 3 years of interest for each of the 274 stations of the network (see Figure \\ref{figSets}), is freely accessible  \\citep[see][fro more details]{MeteoNet}.\n\nLet us note $Y(\\boldsymbol{x})$ the random variable for strictly positive hourly precipitations (dry and wet events are traditionally treated separately when it comes to rainfall simulation \\citep{ailliot2015stochastic}), with covariates $\\boldsymbol{x}=(x_l, x_L, x_t)$, where $x_l$ and $x_L$ represents the position in space (longitude, latitude) and $x_t$ a cyclic time index, namely the day or the month of the year. The work of \\citet{naveau2016modeling} and our tests show that EGPD fitting is improved by censoring the data, so in practice we remove values $Y(\\boldsymbol{x}) < 0.5$ (mm). Other thresholds ($\\leq 0.2$ and $\\leq 0.1$mm) have been tested but $Y(\\boldsymbol{x}) < 0.5$ (mm) gives the  best results, in particular for the upper tails. Besides, as noted in \\citet{evin2018stochastic,naveau2016modeling}, negative shape parameters $\\xi$ are not expected for rainfall distributions at these time scales, so we use a link function constraining the parameter $\\xi$ to positive values. Finally, as performed in \\citet{naveau2016modeling}, we only keep one every three hourly records of precipitation so as to ensure independency between samples.\n\n\\subsection{Model fitting}\n\nWe fit different specifications of EGPD-based models, where EGPD takes form 1, 3 or 4, and compare them with equivalently specified Gamma-based models. The Gamma (GA) distribution is a classical choice that works well for modeling the bulk of precipitations at a given site \\citep{katz1977precipitation,vrac2007stochastic,vlvcek2009daily}. However non-negligible deviations are observed when one is interested in capturing extreme rainfalls \\citep{katz2002statistics}, typically underestimated due to the lightness of the GA tail. For a marginal model  \\texttt{M=EGPD1,EGPD3,EGPD4,GA}, the different specifications read:\n\\begin{description}\n\t\\item \\texttt{M.0} where\n$$\t\\eta_a(\\boldsymbol{x})=\\beta_0 , \\qquad a=\\mu,\\sigma,\\nu,\\tau,$$\ni.e.\t the parameters $\\theta(\\boldsymbol{x})$ do not depend on the covariates ;\n\t\\item  \\texttt{M.t} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ s_t (x_t) , \\qquad a=\\mu,\\sigma,\\nu,\\tau,\n\t$$\n\t\tuses cyclic cubic splines $s_t$ to model the nonlinear dependency in time while considering no variation in space ;\n\t\\item \\texttt{M.tnomu} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ s_t (x_t) , \\qquad a=\\sigma,\\nu,\\tau,\n\t$$ does the same as before but considers as constant the parameter $\\mu$ ;\n\t\\item \\texttt{M.st} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ TP(x_l,x_L)+ s_t (x_t), \\qquad a=\\mu,\\sigma,\\nu,\\tau,\n\t$$\n\tuses smooth surface fitting (thin-plate splines) to model the dependency in space as well as cyclic cubic splines for the dependency in time ;\n\t\\item \\texttt{M.st2mu} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ TP(x_l,x_L)+ s_t (x_t), \\qquad a=\\mu,\\sigma,\n\t$$\n\tdoes the same as before but considers constant in space the parameters that are not $\\mu$ or $\\sigma$ ;\n\t\\item \\texttt{M.st2nomu} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ TP(x_l,x_L)+ s_t (x_t), \\qquad a=\\sigma,\\nu,\n\t$$\n\tdoes the same as before but considers constant in space the parameters $\\mu$ and $\\tau$ ;\n\t\\item \\texttt{M.st3nomu} where \n\t\t\t$$\n\t\\eta_a(\\boldsymbol{x})=\\beta_0+ TP(x_l,x_L)+ s_t (x_t), \\qquad a=\\sigma,\\nu,\\tau,\n\t$$\n\tdoes the same as before but applies to \\texttt{M=EGPD4} only and considers parameter $\\mu$ as a constant over space.\n\\end{description}\n\nThese variations within each model class will allow us to assess whether the smooth regression of distribution parameters over covariates is justified or not. \nBelow we reproduce the code to fit \\texttt{megpd1.0}, \\texttt{megpd1.tnomu} and \\texttt{mga.st} on the training set (\\texttt{training}), that contains 60 \\% of the stations, as reported on Figure \\ref{figSets}. The dataset consisting of the remaining stations form the validation set (\\texttt{validation}). The \\texttt{gamlss()} function reads:\n\n\\begin{lstlisting}[language=R]\n# Notations: lat, lon and cyc represent, respectively, covariates xl, xL and xt, while y is given by precip\n# Set algorithm control parameters:\ncon <- gamlss.control (n.cyc = 200, mu.step = 0.01, sigma.step = 0.01, nu.step = 0.01,tau.step = 0.01, autostep=TRUE)\nmegpd1.0 <- gamlss(precip ~ 1,\n       data = training, \n                  family = EGPD1Family(mu.link = \"own\"),\n                  control = con,\n                  method=CG())         \nmegpd1.tnomu <- gamlss(precip ~ 1,\n                  sigma.fo =~ pbc(cyc),\n                  nu.fo =~ pbc(cyc),\n                  data = training, \n                  family = EGPD1Family(mu.link = \"own\"),\n                  control = con,\n                  method=CG()) \n                      \n# Load library to interface with the mgcv package and get ga() for multidimensional tensor spline smoothers:\nlibrary(gamlss.add)\nfmla_ga <- list(~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50),\n                ~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50))\nGA.st <- gamlss(precip ~ga(~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50),                                          method=\"REML\"), \n                      ~ga(~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50), \n                      method=\"REML\"),\n                      ~ga(~s(lon,lat,bs='tp',k=30)+s(cyc,bs=\"cc\",k = 50), \n                      method=\"REML\"),\n                      data = training, \n                      family = GA,\n                      control = con,\n                      method=CG())\n\\end{lstlisting}\n\n\\subsection{Model assessment}\n\nIn order to evaluate the goodness of fit on the training set, we use summary statistics: Global Deviance (GD), Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).\nThe \\texttt{GAIC(modelName,k=a)} function allows to compute GD (minus twice the fitted log-likelihood, $a=0$), the AIC ($a=2$) and the BIC ($a=\\log(\\mbox{number of observations})$) for each fitted model \\texttt{modelName} and to compare them as performed on Figure \\ref{figInfCriteria}. Besides, Table \\ref{table1} indicates the effective degree of freedom (df) computed for each model. The function call is reproduced below for the models of class EGPD1.\n\n\\begin{lstlisting}[language=R]\nGAIC_BIC_m1 <-GAIC(megpd1.0,megpd1.t,megpd1.tnomu,megpd1.st,megpd1.st2mu,megpd1.st2nomu,k=log(length(training$precip))\n\\end{lstlisting}\n\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.5\\columnwidth]{P3_SituationMapSets.png}\n\t\t\\caption{French stations from the MeteoNet public dataset used for our experiments, split into training and validation sets.\n\t\t\t{\\label{figSets}}\n\t\t}\n\t\\end{center}\n\\end{figure}\n\n\\begin{table}[h]\n\t\\begin{center}\n\t\t\\begin{tabular}{||l c||} \n\t\t\t\\hline\n\t\t\tModel & \\multicolumn{1}{c||}{df} \\\\ [0.5ex] \n\t\t\t\\hline\\hline\n\t\t\t\\texttt{megpd4.st}  & 242.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.st3nomu}  & 199.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.st2nomu}  & 138.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.st2mu}  & 122.5 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.t}  & 72.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.tnomu} & 38.0 \\\\ \n\t\t\t\\hline\n\t\t\t\\texttt{megpd4.0} & 4 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.st}  & 183.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.st2nomu}  & 139.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.st2mu}  & 121.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.t}  & 54.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.tnomu}  & 37.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd1.0}  & 3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.st}  & 181.6 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.st2nomu}  & 137.3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.st2mu}  & 122.9 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.t}  & 54.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.tnomu}  & 37.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{megpd3.0}  & 3 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{mga.st}  & 120.2 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{mga.stnomu}  & 78.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{mga.t}  & 36.0 \\\\\n\t\t\t\\hline\n\t\t\t\\texttt{mga.0}  & 2 \\\\ [1ex] \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{table1} Effective degree of freedom (df) for the fitted models.}\n\\end{table}\n\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.85\\columnwidth]{P4_InfCrit_m1_f.png}\n\t\t\\includegraphics[width=0.85\\columnwidth]{P4_InfCrit_m3f.png}\n\t\t\\includegraphics[width=0.85\\columnwidth]{P4_InfCrit_m4f.png}\n\t\t\\includegraphics[width=0.85\\columnwidth]{P4_InfCrit_gaf.png}\n\t\t\\caption{Values above the global minimum of each information criterion associated with fitted models: global deviance (GD), AIC, BIC on fitting set, and GD on validation set (GD-v). From top to bottom, we compare the variations of EGPD1, EGPD3, EGPD4 and Gamma-based models. The model variations are indicated in full letters on the right side of the points, in an order following their relative ranking.\n\t\t\t{\\label{figInfCriteria}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\nFrom Figure \\ref{figInfCriteria}, we can see that, as could be expected, for each classes of models (where EGPD1-based models represent one class, EGPD3-based ones another, etc) the GD ranks first models with the full space-time description \\texttt{M.st}. All other variations are rather close in fitting performance, at the exception of the constant-parameters one \\texttt{M.0}, that is largely behind. This shows the clear interest of allowing a smooth variation of distribution parameters as a function of the chosen covariates. However, when it comes to modeling efficiency, where the trade-off between number of parameters and performances is taken into account (through the AIC and BIC), the full space-time model \\texttt{M.st} becomes much less desirable, falling just  above and occasionally behind the constant parameter models \\texttt{M.0}. This is all the more through that we consider the BIC indice, that penalizes more the number of parameters than the AIC, and a model with a large number of parameters (EGPD4 versus GA). The best trade-off, according to the BIC, are the time-only full model \\texttt{M.t} for the GA class and the time-only without the shape parameter $\\mu$ otherwise. This may also be an indication that the seasonal variation of distribution parameters is more important than their spatial variation over the area of interest. \n\nThe ranking of model variations within a same model class allows us to draw conclusions w.r.t. the advantages or challenges of each class. Thus, for EGPD4 the GD ranks model variations following decreasing complexity: \\texttt{megpd4.st} followed by \\texttt{megpd4.st3nomu}, etc. We can deduce that the estimation of $\\mu$ is not that sensitive when it comes to the fitting set (at least with enough data). On the contrary, \\texttt{megpd1.st2nomu} is preferred to \\texttt{megpd1.st} and \\texttt{megpd1.st2mu} which can be interpreted as $\\mu$ being a rather sensitive parameter to estimate for EGPD1. Considering it constant in space consequently improves the global fitting. \n\nPerformances on the validation set are obtained by means of the \\texttt{getTGD()} and \\texttt{TGD()} functions, providing the validation global deviance (GD-v), that is the GD evaluated using predictive values for the parameters at the validation sample.\n\\begin{lstlisting}[language=R]\n\tgg.megpd4.st <- getTGD(megpd4.st,newdata=validation)\n\tgg.megpd4.0 <- getTGD(megpd4.0,newdata=validation)\n\t...\n\tTGD(gg.megpd4.0,gg.megpd4.t,gg.megpd4.tnomu,gg.megpd4.st,gg.megpd4.st2mu,gg.megpd4.st2nomu,gg.megpd4.st3nomu)\n\\end{lstlisting}\nThus, when it comes to extrapolating parameters for new covariate values, the GD-v ranking displayed in Figure \\ref{figInfCriteria} reveals that models considering only the dependence in time (\\texttt{M.t}, \\texttt{M.tnomu}) are systematically preferred, followed by space-time models not considering $\\mu$ (\\texttt{M.st2nomu}), before the full space-time models (\\texttt{M.st}). This supports the fact that the rainfall distributions are relatively close over the studied geographical area, compared to their time evolution that shows stronger variations. Consequently, given the limited size of the dataset, keeping low the number of parameters to regress on covariates (by considering the seasonality only) and in particular avoiding in this category the shape parameter $\\mu$, makes estimations more robust. This agrees with observations from \\citet{evin2018stochastic}, who noted that fitting the shape parameter is the most challenging part of (extended)-GPD modeling. They concluded that mono-site fitting is not robust, leading e.g. to negative shape parameters that do not match observations on rainfall intensities. To circumvent this issue, the authors use constant parameters within regions of influence defined by homogeneity tests.\n\n\\subsubsection{Model verification and comparison}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\columnwidth]{P4_Uplot_w.png}\n\t\t\\includegraphics[width=0.48\\columnwidth]{P4_Uplot_sp.png}\n\t\t\\includegraphics[width=0.48\\columnwidth]{P4_Uplot_su.png}\n\t\t\\includegraphics[width=0.48\\columnwidth]{P4_Uplot_au.png}\n\t\t\\caption{Assessment of the quality of each model (EGPD1, EGPD3, EGPD4 and GA) for all stations from the verification set in (from left to right and top to bottom) winter, spring, summer and autumn. The closer to the diagonal, the better the underlying model distribution. See text for theoretical development. We compare the \\texttt{st2nomu} version of each model class (that is where parameters $\\mu,\\tau$ are constant in space). Theoretical probabilities $\\leq 0.4$ is overall equivalent to $y \\leq 1$ mm where $y$ designates the hourly rainfall amount.\n\t\t\t{\\label{figUplots}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\n\nNow that it is admitted that regressing distributional parameters over time and possibly space is justified for the case study at hand, we want to assess how good is the modeling and compare model classes at hand.\nA way to verify whether the fitted models actually represents well the empirical distribution is to draw so-called P-P plots. \nA simple result in probability states if $Y(\\mathbf{x}) \\sim F(\\cdot;\\theta(\\mathbf{x})$ then $U(\\mathbf{x}):=F(Y(\\mathbf{x});\\theta(\\mathbf{x}))$\nis uniformly distributed on the unit interval. \n\nTherefore we can consider the  residuals $u(\\mathbf{x}) = F(y(\\mathbf{x}),\\widehat{\\theta}(\\mathbf{x}))$ where $\\widehat{\\theta}(\\mathbf{x})$ are the parameters\n of the fitted model and, after having ordered them,  compare them with  $\\Big( \\frac{1}{n+1}, ..., \\frac{n}{n+1} \\Big)$. Here $n$ is the number of observations $y(\\mathbf{x})$ at hand. The closer points are to the diagonal, the better the model fit. Results are broken down at each season for model variations \\texttt{M.st2nomu} on the verification set in Figure \\ref{figUplots}. \nThey show that at all seasons, EGPD4 outperforms the other models. This is all the more true that we consider not too small rainfalls, namely hourly amounts $x \\geq 1$ mm. Below, we keep wiggly artefacts of the discrete sampling scale of observations. Indeed, rainfall amounts are sampled with a precision of $0.2$ mm, which makes the distribution of small values more discrete than continuous. Behind, come models GA, EGPD3 and finally EGPD1. EGPD3 is often better than GA for small hourly rain amounts (below around $2$ mm) however it may be outperformed for larger ones, especially in winter. EGPD1 performs as EGPD3 for intermediate and large amounts of rain, but is significantly behind all other models for small amounts.\n\n\\subsubsection{Visualisation of the fitted functions}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.75\\columnwidth]{P3_parEvol_statNN.png}\n\t\t\\includegraphics[width=0.75\\columnwidth]{P3_ParEvol_stationS.png}\n\t\t\\caption{EGPD4 coefficients fitted for the models \\texttt{megpd4.0} (constant parameters), \\texttt{megpd4.t} (smooth time dependence only), and \\texttt{megpd4.st} (full smooth space time dependence) in two stations: (top two rows) on the north-west coast of Brittany ($x_L=48.7$ $x_l=-4.33$) and (bottom two rows) in the south-east countryside of the studied area ($x_L=46.7$ $x_l=1.25$). Coefficients are fitted against the covariate $cyc$, namely time of the year. The shadowed areas correspond to the associated standard errors.\n\t\t\t{\\label{figParEvol}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\n\nFigure \\ref{figParEvol} represents the fitted functions for the variations \\texttt{megpd4.0} (constant parameters), \\texttt{megpd4.t} (smooth time dependence only), and \\texttt{megpd4.st} (full smooth space time dependence) of the EGPD4 model in two \\textit{a priori} rather different stations when it comes to the rainfall regime: the north-west coast of Brittany ($x_L=48.7$ $x_l=-4.33$) and the south-east countryside of our area of interest ($x_L=46.7$ $x_l=1.25$).\n \nThe functions \\texttt{predict()} or \\texttt{predictAll()} are used to provide fitted values of the functions of the covariates. Standard errors can be extracted for fitted values by means of the argument \\texttt{se.fit=TRUE}. However this option is not supported by \\texttt{gamlss} for parameter estimates on new covariate values at the moment of writing this paper. \n\\begin{lstlisting}[language=R]\n# Fitted parameter values:\nmuFit <- predict(megpd4.st,what=\"mu\",type=\"response\")\n\n# Estimates for new covariate values, gathered in the database \"validation\":\nall <- predictAll(megpd4.st,data=training,newdata=validation)\nmuVal <- all$mu\nsigmaVal <- all$sigma\nnuVal <- all$nu\n\nmuFit <- predict(megpd4.st,what=\"mu\",type=\"response\",data=training,se.fit=TRUE)\nStandardErrorMu <- muFit$se.fit\n\\end{lstlisting}\nWe observe rather similar temporal behaviors of the parameters estimated in both stations, highlighting the relative spatial homogeneity of distributions compared to their seasonal dependence. The standard errors decrease with the simplicity of the model ; they are generally negligible for for parameters $\\sigma$ and $\\tau$ but can become significant for parameters $\\nu$ and $\\mu$ when we are dealing with space-time models. Beyond that, the estimations for the three different versions of the EGPD4-based model are consistent. \nWe note (second station) that a shift between \\texttt{megpd4.t} and \\texttt{megpd4.st}'s estimates of parameter $\\sigma$ is compensated by an opposite shift between the associated estimates of parameter $\\tau$.\nFrom our observations and experience on EGPD fitting, a given EGPD-based distribution is matched by several local optima (i.e. parameter combinations). Along this \"Pareto front\", $\\nu$ is increasing for EGPD1 when $\\sigma$ decreases. Similarly  $\\tau$ is increasing for EGPD4 when $\\sigma$ decreases. \n\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\includegraphics[width=\\columnwidth]{P3_egp4st_12.png}\n\t\t\\includegraphics[width=\\columnwidth]{P3_egp4st_4.png}\n\t\t\\includegraphics[width=\\columnwidth]{P3_egp4st_8.png}\n\t\t\\caption{Median parameter maps for the full EGPD4 model \\texttt{megpd4.st} for the months of December (top row), April (middle row) and August (last row).\n\t\t\t{\\label{figParMaps}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\n\nFinally, Figure \\ref{figParMaps} represents the spatial distribution of the median of the EGPD4 parameters estimated from \\texttt{megpd4.st} at three different times of the year: december, april and august. We indeed observe spatial variations of the parameter values, in addition to the monthly changes already discussed. $\\mu$ the most spatially homogeneous parameter, which goes in line with previous observations that favored spatially constant $\\mu$-models over non-constant ones.\nWe also note that the $\\tau$ map follows an opposite trend w.r.t. $\\sigma$'s map: the larger $\\tau$, the smaller $\\sigma$, which follows the general idea of parameter Pareto front developed earlier.\n\n\n\\subsubsection{Goodness-of-fit w.r.t. the upper tail}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext099_w.png}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext0995_w.png} \\\\\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext099_sp.png}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext0995_sp.png} \\\\\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext099_su.png}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext0995_su.png} \\\\\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext099_au.png}\n\t\t\\includegraphics[width=0.3\\columnwidth]{P4_Uplots_ext0995_au.png}\n\t\t\\caption{Assessment of the quality of model EGPD4's variations and GA for all stations from the validation set in (from top to bottom) winter, spring, summer and autumn. The closer to the diagonal, the better the underlying model distribution, see Figure \\ref{figUplots}. We focus on extremes, with theoretical probabilities $\\geq 0.99$ (left column) equivalent to hourly rainfall amounts above $11$ mm for EGPD4 models and above $7$ mm for GA. Theoretical probabilities $\\geq 0.995$ (right column) are equivalent to rain amounts above $15$ mm for EGPD4 models and above $7.6$ mm for GA.\n\t\t{\\label{figUplotext}}%\n\t\t}\n\t\\end{center}\n\\end{figure}\nWe finish the presentation of the results by comparing in Figure \\ref{figUplotext} the EGPD4 model to a GA fit, for all stations of the validation set and for each season. We  want to check the added value of the EGPD4 model over the GA when it comes to extremes, as well as to assess the best modeling option for the EGPD4 shape parameter in order to ensure a correct representation of extremes.\nThe series of P-P plots presented on Figure \\ref{figUplotext} focus on the upper tail of the distribution (above the quantiles 0.99 and 0.995 respectively) and allow us to do that. We first note that whatever the season, EGPD4 performs significantly better than GA when it comes to extremes. Most often, the constant parameter model \\texttt{megpd4.0} is enough to outperform the GA, and sometimes is clearly the best option as in spring and autumn. In winter and summer, it is not enough to capture correctly the upper tail of the empirical distribution. Making EGPD4 parameters dependent on time only is sufficient to obtain correct upper tail representations, whether $\\mu$ is considered constant (\\texttt{megpd4.tnomu}) or not (\\texttt{megpd4.t}). The difference between the last two models is very small however using the constant shape parameter option can slightly improve results, especially for larger quantiles (0.995 and above versus 0.99). Adding spatial dependence to the temporal one improves even more the accuracy of the EGPD4 upper tail modeling. Making spatially dependent only two parameters usually performs better. However $\\mu$ needs to be included in these two parameters (\\texttt{megpd4.st2} versus \\texttt{megpd4.st2nomu}). \nThis highlights the fact that a smooth space-time estimation of the shape parameter $\\mu$ is important to capture correctly the non-stationary behavior of the upper tail  of the rainfall distributions. However, due to limited data and numerical sensitivity of that estimation, it may be preferable to only limit the smooth spatial dependence to two parameters instead of four, including $\\mu$, in order to make it more robust for extrapolation.\n\n\\section{Conclusions}\n\\label{secConclu}\n\nThis article introduced an add-on to the R-package \\texttt{gamlss}, to model non-stationary fields whose margins are EGPD distributed. A number of R packages target the modeling of heavy-tailed distributions. Yet, although some allow distributional regression models, for which distribution parameters can be modeled as additive functions of covariates, none of these packages address the extended-GPD distribution nor another form of unique and closed-form formulation for heavy-tailed distributions that models small, intermediate and large values in a synthetic manner. This add-on implements the EGPD family in a generic way in the \\texttt{gamlss} package, allowing to test any new parametric form, up to 4 parameters so far, satisfying the requirement of the EGPD family. \nApplications include space-time modeling of rainfall marginal distributions. The example developed in this paper shows 1) that modeling hourly rainfall amounts is significantly improved when distribution parameters are non-stationary in space and time; and that 2) the EGPD class clearly improves the modeling of intermediate values and upper tail compared to a classical Gamma option. This illustrates the benefits of our add-on with respect to existing GAM forms for modeling non-stationary margins (typically the Gamma model for our application). We compare three parametric EGPD models and discuss the modeling strategies to best capture non stationary upper tails with limited datasets. This case study also reveals the added-value of EGPD4 (and to a less extent EGPD3) over EGPD1 to correctly model hourly rainfall distributions in a space-time context, while previous works \\cite{naveau2016modeling,evin2018stochastic,bertolacci2018comparison} restricted themselves to EGPD1 as the best option.\n\n\n\\section*{Software and data availability}\n\nThe add-on to the R-package \\texttt{gamlss} \\citep{gamlssP} presented here for the generic implementation of EGPD consists in the script \\texttt{GenericEGPDg.R}, available at \\url{https:\/\/github.com\/noemielc\/egpd4gamlss}. On this same webpage, supplementary material consisting in a reproducible tutorial with code is presented.\n\nThe data used in this paper is freely accessible on the MeteoNet French database \\citep{MeteoNet}: \\url{https:\/\/meteofrance.github.io\/meteonet\/english\/data\/summary\/}\n\n\\section*{Acknowledgement}\n\nScientific activity was performed as part of the research program Venezia2021, coordinated by CORILA, with the contribution of the Provveditorato for the Public Works of Veneto, Trentino Alto Adige and Friuli Venezia Giulia (Italy).\nThe authors thank Nicolas Berthier for helpful R suggestions.\n\n\n\\bibliographystyle{elsarticle-harv.bst}\n\n\t\\setlength\\baselineskip{.96\\baselineskip}%\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}