diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgtxe" "b/data_all_eng_slimpj/shuffled/split2/finalzzgtxe" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgtxe" @@ -0,0 +1,5 @@ +{"text":"\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nSpeyer \\cite{Speyer:2016:VOATOKW} recently published a very short and elegant proof\nfor Kasteleyn's Theorem \\cite{Kasteleyn:1967:GTACP}. This proof involves some higher topology arguments; and\nthe purpose of this note is to replace them by elementary graph--theoretical\narguments.\nWe shall, however, present only a sketch of a proof, since the technical details\nof a fully rigorous proof would likely conceal the beauty and simplicity of the basic idea.\n\n\n\\section{Basic definitions and notation}\n\nFor reader's convenience and to fix our notation, we shall briefly\nrecall some basic definitions.\n\n\\subsubsection*{Graph}\n\nA finite graph $G$ consists\n\\begin{itemize}\n\\item of a finite set of\n\\EM{vertices} $V\\of G$\n\\item and a finite (multi--)set of \\EM{edges} $E\\of G$,\n\\end{itemize}\nwhere each $e\\in E\\of G$\n\\EM{joins} two different vertices $v_1, v_2\\in V\\of G$ (we also say that $e$ is \\EM{incident}\nwith $v_1$ and $v_2$), which we denote by $e=\\setof{v_1,v_2}$.\nWe shall not consider \\EM{loops}, which are\nedges ``joining a single vertex'' ($e=\\setof{v}$): If such loop arises during the constructions\ndescribed below, we shall remove it immediately.\n\n\\subsubsection*{Graph embedding}\n\nA finite graph $G$ can always be \\EM{drawn} in the plane ${\\mathbb R}^2$, such that\n\\begin{itemize} \n\\item the vertices correspond bijectively to \\EM{points} in the plane (we shall call them \\EM{vertex--points}),\n\\item and the edges correspond bijectively to smooth \\EM{curves} $\\brk{0,1}\\to{\\mathbb R}^2$ in the plane (we shall call them \\EM{edge--curves})\n\twhich \\EM{connect} the respective vertex--point, i.e.,\n\tsome edge $e=\\setof{v_1,v_2}$ corresponds to an edge--curve\n\t\\begin{itemize}\n\t\\item which starts in the vertex--point corresponding to $v_1$\n\tand ends in the vertex--point corresponding to $v_2$ (or vice versa: the orientation of the curve is irrelevant in our context),\n\t\\item but \\EM{does not} ``intersect itself'' (i.e., does not pass twice through the same point;\n\tin particular, edge--curves do not ``return'' to their starting point or end point),\n\t\\item and does not pass through any other vertex--point.\n\t\\end{itemize}\n\\end{itemize}\nSuch drawing is called an \\EM{embedding} of the graph $G$ in the plane.\n\nBy definition, the edge--curves of edges incident to the same vertex $v$ have\nthe vertex--point corresponding to $v$ in common,\nbut there might be other such intersections in \\EM{nonvertex--points}.\n\nFor our purpose, we additionally assume that:\n\\begin{enumerate}\n \n \n\n\\item Any two curves intersecting in some nonvertex--point $p$\n\tdo actually \\EM{cross} each other (i.e., they are \\EM{not tangent} in $p$),\n\twhence we call such point $p$ a \\EM{crossing} of the two curves.\n\\item There is only a finite number of such \\EM{crossings}.\n\\end{enumerate}\nWe call an embedding which fulfils these additional assumptions a \\EM{proper embedding}.\n\nSee \\figref{fig:simple-graph} for an\nillustration of these concepts: In all graphical representations of proper embeddings in this\nnote, we shall indicate\n\\begin{itemize}\n\\item vertex--points by \\EM{black dots},\n\\item curves corresponding to edges by \\EM{gray lines},\n\\item and crossings by \\EM{small white circles}.\n\\end{itemize}\n\\begin{figure}\n\\label{fig:simple-graph}\n\\caption{Example of a proper embedding of some graph $G$ in the plane.}\n{\\small\nLet $V\\of G = \\setof{v_1,v_2,\\dots, v_6}$ and $E\\of G = \\setof{\n\\setof{v_1,v_2},\n\\setof{v_2,v_3}^2,\n\\setof{v_3,v_4},\n\\setof{v_5,v_6}}$, where the notation $\\setof{v_2,v_3}^2$ indicates that there are\n\\EM{two} edges joining vertices $v_2$ and $v_3$. The picture shows a proper embedding of $G$\nwith a single crossing $p$: The points corresponding to the vertices are indicated by black\ndots, the curves corresponding to edges are indicated by gray lines, and the crossing is indicated by a small\nwhite circle.\n}\n\\begin{center}\n\\input graphics\/simple-graph.tex\n\\end{center}\n\\end{figure}\n\n\\subsubsection*{Simplified notation}\nIf we are given some graph $G$ and an embedding $\\eta$ of $G$, let us denote\n\\begin{itemize}\n\\item by $\\eta\\of v$ the vertex--point representing vertex $v\\in V\\of G$,\n\\item by $\\eta\\of e$ the edge--curve representing edge $e\\in E\\of G$.\n\\end{itemize}\nBut we shall abuse this notation in the following by making no distinction\n\\begin{itemize}\n\\item between $v\\in V\\of G$ and $\\eta\\of v$, i.e., instead of saying\n\t``the vertex--point in the plane representing $v$'' we simply shall say ``$v$'',\n\\item and between $e\\in E\\of G$ and $\\eta\\of e$, i.e., instead of saying\n\t``the edge--curve in the plane representing $e$'' we simply shall say ``$e$''.\n\\end{itemize}\nThis slight imprecision should cause no confusion in our context.\n\n\\subsubsection*{Planar graph}\n\nA proper embedding without any intersection of edge--curves in non--vertex points is called a\n\\EM{planar embedding}.\nA graph which admits a planar embedding is called a \\EM{planar graph}.\n\nNot all graphs are planar, but we may \\EM{view} every proper embedding of some\ngraph $G$ as planar embedding of a planar graph $G^\\prime$ by \\EM{reinterpreting} all\ncrossings as vertex--points: $G=G^\\prime$ (in the sense of graph isomorphisms) if the\nembedding has no crossings at all. (In \\figref{fig:simple-graph}, this reinterpretation would amount\nto replacing the single crossing $p$ by a new vertex--point $v_7$.)\n\n\\subsubsection*{Perfect matching}\n\nA \\EM{perfect matching} $M\\subseteq E\\of G$ of $G$ is a subset of edges\nsuch that every vertex of $G$ is incident with precisely one edge in $M$.\n\n\\subsubsection*{Edge weight}\n\nWe assume that we are given some (nowhere--zero) \\EM{weight function} (called \\EM{edge weight}) $\\omega$\n$$\\omega:E\\of G\\to R\\setminus\\setof{0}$$\non the multiset of edges of $G$, where $R$ is some (nontrivial) commutative\nring (in most cases, $R$ is ${\\mathbb Z}$ or some polynomial ring; the constant edge weight \n$\\omega\\equiv 1$ is used for enumeration purposes).\n\n\n\\subsubsection*{Generating function of perfect matchings}\n\nThe weight of some perfect matching $M\\subseteq E\\of G$ is the product of the weights of the edges in $M$:\n$$\n\\omega\\of M:=\\prod_{e\\in M} \\omega\\of e.\n$$\n\nThe \\EM{generating function} of perfect matchings of some graph $G$ with edge weight $\\omega$ is\ndefined as\n$$\nm\\of{G,\\omega}:=\\sum_{M}\\omega\\of M,\n$$\nwhere $M$ ranges over all perfect matchings of $G$.\n\n\\subsubsection*{Sign of perfect matchings}\n\\def\\operatorname{img}{\\operatorname{img}}\nAssuming a fixed proper embedding $\\eta$ of graph $G$, the \\EM{sign} of some perfect matching\n$M$ of $G$ in the embedding $\\eta$ is defined as\n$$\n\\sgn\\of{M,\\eta}:=\\pas{-1}^{C\\of{M,\\eta}},\n$$\nwhere $C\\of{M,\\eta}$ is the \\EM{number of all crossings of all edges} in $M$,\ni.e., denoting by $\\operatorname{img}\\of{e}$ the image of the (edge--curve corresponding to) edge $e$:\n\\begin{equation}\n\\label{eq:crossing}\nC\\of{M,\\eta}:=\\sum_{\\setof{e_1,e_2}\\subseteq M} \\absof{\\operatorname{img}\\of{e_1}\\cap\\operatorname{img}\\of{e_2}}.\n\\end{equation}\n\nIt is possible to modify some proper embedding $\\eta$ to another embedding $\\eta^\\prime$ so that\nthe signs of all perfect matchings are the same for $\\eta$ and $\\eta^\\prime$: See \\figref{fig:crossing}\nfor a simple example of such \\EM{sign--preserving} modification.\n\n\\subsubsection*{Signed generating function of perfect matchings}\n\nThe \\EM{signed generating function} of perfect matchings\n$s\\of{G,\\omega,\\eta}$ (which depends not only on the graph $G$, but also on the proper embedding $\\eta$)\nis defined as\n$$\ns\\of{G,\\omega,\\eta}:=\\sum_{M}\\sgn\\of{M,\\eta}\\cdot\\omega\\of M,\n$$\nwhere $M$ ranges over all perfect matchings of $G$.\n\nClearly, $m\\of{G,\\omega}=s\\of{G,\\omega,\\eta}\\equiv 0$ if\n$\\absof{V\\of G}$ is {odd}.\n\n\n\\begin{figure}\n\\caption{Untangling of a multiple crossing.}\n\\label{fig:crossing}\n{\\small\nThe left picture shows \\EM{four} edge--curves passing through the \\EM{same} nonvertex--point: \nObserve that this situation can be ``modified locally'' (in the sense that the embedding is left\nunchanged outside some small neighbourhood of the crossing) to a situation like the one\nshown in the right picture, and that such ``local modification'' does not change the sign\nof any perfect matching. For instance, if we assume that the edges\nshown here belong to some perfect\nmatching $M$, both ``local situations'' contribute a factor $\\pas{-1}$ to the sign of $M$\nfor \\EM{every two--element subset} of the four curves (i.e., $\\pas{-1}^{\\binom{4}2}=\\pas{-1}^6=1$).\n}\n\\begin{center}\n\\input graphics\/star.tex\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection*{Stembridge's embedding and the Pfaffian}\nFor every graph $G$ with vertex set $V\\of G=\\setof{v_1,v_2,\\dots,v_n}$\nthere is a specific proper embedding which we shall call \\EM{Stembridge's embedding}\n(see \\cite{Stembridge:1990:NPPAPP}):\nEvery edge $e=\\setof{v_i,v_j}$, $i\\neq j$, is represented by the half--circle in the upper half--plane\nwith center $\\pas{\\frac{i+j}2,0}$ and radius $\\frac{\\absof{i-j}}2$,\nand every vertex $v_i$ is represented by the point $\\pas{i,0}$.\n(See the left picture in \\figref{fig:pfaffian} for an example of Stembridge's embedding.)\n\nDenote this specific embedding by $\\overline{\\eta}$: The \\EM{Pfaffian} of a graph $G$ is defined\nas the signed generating function for the embedding $\\overline{\\eta}$, i.e.:\n$$\n\\operatorname{Pf}\\of{G,\\omega}:= s\\of{G,\\omega,\\overline{\\eta}}.\n$$\n\n\\begin{figure}\n\\caption{Stembridge's embedding and the Pfaffian.}\n\\label{fig:pfaffian}\n{\\small\nThe left picture shows Stembridge's embedding $\\overline{\\eta}$ of a graph $G$.\n\nThe right picture shows a specific perfect matching $M$ of $G$ for which the number $C\\of{M,\\overline{\\eta}}$\nof crossings according to \\eqref{eq:crossing} equals $3$, whence the sign\n$\\sgn\\of{M,\\overline{\\eta}}$ of this matching is $\\pas{-1}^3=\\pas{-1}$. \n}\n\\begin{center}\n\\input graphics\/pfaffians.tex\n\\end{center}\n\\end{figure}\n\n\\subsubsection*{Sign--modifications of edge weights and Kasteleyn's Theorem}\nLet $G$ be some simple graph with edge weight $\\omega$: Another edge weight $\\omega^\\prime$\nis called a \\EM{sign--modification} of $\\omega$ if for all $e\\in E\\of G$ there holds\n$$\n\\omega^\\prime\\of e = \\omega\\of e \\text{ or }\\omega^\\prime\\of e = -\\omega\\of e.\n$$\n\nThen we may formulate Kasteleyn's Theorem \\cite{Kasteleyn:1967:GTACP} as follows:\n\n\\begin{thm}[Kasteleyn's Theorem]\n\\label{thm:kasteleyn}\nLet $G$ be a \\EM{planar} finite simple graph\nwith edge weight $\\omega$. Let $\\eta$ be an arbitrary proper\nembedding of $G$. Then there exists a sign--modification $\\omega^\\prime$\nof $\\omega$ such that\n$$\nm\\of{G,\\omega} =\n\\operatorname{Pf}\\of{G,\\omega^\\prime}.\n$$\n\\end{thm}\n\nNote that for any \\EM{planar} embedding $\\eta$ of $G$ we have $m\\of{G,\\omega}=s\\of{G,\\omega,\\eta}$.\n\nWe will rephrase Speyer's elegant argument \\cite{Speyer:2016:VOATOKW} to\ngive a sketch of proof for the following slight generalization:\n\\begin{thm}\n\\label{thm:kasteleyn2}\nLet $G$ be a finite simple graph\nwith edge weight $\\omega$. Let $\\eta$ be a\nproper embedding of $G$.\nThen for \\EM{every} proper embedding $\\pas{G,\\eta^\\prime}$ \nthere is a sign--modification $\\omega^\\prime$ of $\\omega$ such that\n$$\ns\\of{G,\\omega,\\eta}\\equiv s\\of{G,\\omega^\\prime,\\eta^\\prime}.\n$$\n\\end{thm}\n\n\\section{Speyer's argument, in simple pictures}\nThe simple idea for the proof of Theorem~\\ref{thm:kasteleyn2} is to \\EM{successively transform} the\nembedding $\\eta$\n\\begin{itemize}\n\\item by \\EM{local modifications of the embedding}\n\\item and \\EM{corresponding modifications of the edge weight} (if necessary)\n\\end{itemize}\nsuch that the signed generating function of perfect matchings stays \\EM{unchanged},\nuntil the embedding $\\eta^\\prime$ is obtained.\n\nNote that smooth bijections ${\\mathbb R}^2\\to{\\mathbb R}^2$ will transform the embedding without changing the (numbers\nof) crossings, thus leaving the signed generating function of perfect matchings unchanged.\nMore informally: If we think of the embedding\nas a ``web of infinitely thin and ductile strings'' (corresponding to the edges) glued together\nat their endpoints (corresponding to the vertices), then we can imagine that we may ``deform and\ndrag around'' these strings, and such transformation will not change the signed generating function of\nperfect matchings if we do not \\EM{remove} crossings or \\EM{introduce} new ones.\n\nMoreover, there are modifications which leave the sign of every perfect matching unchanged,\nbut cannot be achieved by a continuous mapping ${\\mathbb R}^2\\to{\\mathbb R}^2$: One example is the ``untangling of\na multiple crossing''\nillustrated in \\figref{fig:crossing}.\n\nIn the following, we shall consider ``local modifications of proper embeddings''. For simplicity, we assume that\nevery crossing\nis the intersection of \\EM{precisely two} edge--curves\n(which we can achieve by the ``untangling of\na multiple crossing'' illustrated in \\figref{fig:crossing}; without changing the signed generating function\nwe are interested in).\n\n\\subsubsection*{Crossing edges incident to the the same vertex}\nIf two edges $e_1$, $e_2$ have a vertex $v$ in common, then they can never both belong to\nthe \\EM{same} perfect matching, hence crossings of $e_1$, $e_2$ are\nirrelevant for the sign of any perfect matching. If the curve--segments between\n$v$ and some crossing $p$ do not contain any other crossing, then $p$\ncan simply be removed (or introduced)\nby the modification illustrated in the following picture:\n\\begin{center}\n\\input graphics\/remove-intersection-neighbours\n\\end{center}\nThis modification has no effect on the sign of any perfect matching. If we modify the situation\n``from left to right'', we shall call this \\EM{straightening out a single crossing} of $e_1$ with $e_2$.\n\n\\subsubsection*{Self--crossing edges}\nWe ruled out self--intersections of edge--curves for proper embeddings, but they might arise by one\nof the modifications described below. The following picture makes clear that we can ``change the\nsituation locally'' and remove (or introduce) such self--crossing.\n\\begin{center}\n\\input graphics\/remove-selfintersection\n\\end{center}\nThis modification has no effect on the sign of any perfect matching, since only crossings of \\EM{different}\nedges contribute to the sign.\n If we modify the situation\n``from left to right'', we shall call this \\EM{straightening out a single self--crossing} of $e$.\n\n\\subsubsection*{Dragging a segment of an edge--curve over another segment}\nAssume that segments of edges\n$e$, $e^\\prime$ can be ``dragged over one another''\nsuch that precisely two crossings arise (or vanish), as the following picture illustrates:\n\\begin{center}\n\\input graphics\/edge_over_edge\n\\end{center}\nThis modification has no effect on the sign of \\EM{any} perfect matching.\n If we modify the situation\n``from left to right'', we shall call this \\EM{straightening out a double crossing} of $e$ with $e^\\prime$.\n\n\\subsubsection*{Transition of a vertex through an edge}\nAssume that a segment of\nsome edge $e$ can be ``dragged over some vertex $v$''\nsuch that precisely one crossing arises (or vanishes) for every edge incident with $v$,\nas the following picture illustrates:\n\\begin{center}\n\\input graphics\/edge_over_vertex\n\\end{center}\nSince every perfect matching $M$ must contain \\EM{precisely one} edge incident to $v$, the sign of $M$\nis \\EM{reversed} by this modification if and only if $e\\in M$: But we can easily offset this by replacing\n$\\omega\\of e$ by $-\\omega\\of e$.\n\n\n\\section{Sketch of proof by example}\n\\label{sec:example}\n\nA rigorous proof of Theorem~\\ref{thm:kasteleyn2} \nwould involve tedious technical details: We shall avoid them by \nonly illustrating the \\EM{idea of proof}, i.e, the ``successive modification of embeddings'', \nin an example.\n\nThe example we shall consider is the \\EM{complete bipartite graph} $K_{3,3}$: Three\ndifferent proper embeddings of this graph are shown in \\figref{fig:proper}.\n\nObserve that the embedding shown in the middle of \\figref{fig:proper} can easily\nbe transformed to the embedding\nshown in the right by the ``untangling of a multiple crossing''\nillustrated in \\figref{fig:crossing}.\n\n\\figref{fig:example} illustrates the process of successively modifying the embedding $\\eta$\nshown in the left picture of \\figref{fig:proper} until the embedding $\\eta^\\prime$ shown in the middle\npicture of \\figref{fig:proper} is obtained.\nWe shall explain the single steps in the following, where we denote the edge connecting\nvertices $v_i$ and $v_j$ by $e_{i,j}$.\n\n\\begin{figure}\n\\caption{Three different embeddings of the same graph $G$.}\n\\label{fig:proper}\n{\\small\nThe graph $G$ we are considering here is the \\EM{complete bipartite graph} $K_{3,3}$:\nThe fact that $K_{3,3}$ is \\EM{not planar} corresponds to the fact that we are\nunable to present an embedding without crossings.\n}\n\\begin{center}\n\\input graphics\/proper.tex\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Example: Succesive modification of embeddings.}\n\\label{fig:example}\n{\\small\nConsider the embeddings of the graph $G$ depicted in \\protect{\\figref{fig:proper}}:\nWe want to transform the embedding $\\eta$ shown in the left picture to the embedding $\\eta^\\prime$\nshown in the middle picture of \\protect{\\figref{fig:proper}}.\n}\n\\begin{center}\n\\input graphics\/example.tex\n\\end{center}\n\\end{figure}\n\n\nThe left upper picture in \\figref{fig:example} shows the result of a smooth deformation of the plane,\nwhere vertices $v_2$, $v_4$ and $v_6$ already have arrived at their desired positions\n(according to the hexagonal configuration in $\\eta^\\prime$). Now we want to drag $v_5$ to its \ndesired position along the path indicated by the dashed arrow. This involves a transition\nof $v_5$ through edge $e_{3,6}$, whose weight must therefore change its sign.\n\nThe upper picture in the middle shows the result of this operation: Note that we may now\nstraighten out the double crossing of $e_{4,5}$ with $e_{3,6}$ and the single\ncrossing of $e_{5,6}$ with $e_{3,6}$.\n\nNow we want to drag $v_1$ to its \ndesired position along the path indicated by the dashed arrow. This involves a transition\nof $v_1$ through edge $e_{2,3}$, whose weight must therefore change its sign. Note that after performing\nthis operation, we may straighten out the double crossing of\n$e_{1,4}$ with $e_{2,3}$.\n\nThe right upper picture shows the result of these operations: Observe that we may now straighten\nout the double crossing of $e_{2,3}$ with $e_{1,6}$ and the single crossing of $e_{1,2}$ with $e_{2,3}$.\n\nNow we want to drag $v_3$ to its \ndesired position along the path indicated by the dashed arrow: This can be done without introducing\nor removing any crossing.\n\nThe left lower picture shows the result of these operations. By dragging $e_{1,6}$ over vertex\n$v_4$ and straightening out the single crossing with $e_{1,4}$ thus introduced, we arrive at the lower\npicture in the middle: This involves a transition\nof $v_4$ through $e_{1,6}$, whose weight must therefore change its sign. \n\nNow dragging $e_{3,6}$ over vertices $v_4$ \\EM{and} $v_5$ (and changing the sign of $\\omega\\of{e_{3,6}}$\n\\EM{twice}, accordingly), and straightening out\n\\begin{itemize}\n\\item the two single crossings of $e_{3,6}$ with $e_{5,6}$ and $e_{3,4}$\n\\item and the double crossing of $e_{3,6}$ with $e_{4,5}$\n\\end{itemize}\nintroduced by this modification gives the right lower picture. \n\nRepeating the last step by dragging $e_{2,5}$ over vertices $v_3$ and $v_4$, we arrive at the right proper drawing\nin {\\figref{fig:proper}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\@startsection {section}{1}{\\z@}{-3.5ex plus -1ex minus \n-.2ex}{2.3ex plus .2ex}{\\normalsize\\bf}}\n\\makeatother\n\n\\makeatletter\n\\def\\subsection{\\@startsection {subsection}{1}{\\z@}{-3.5ex plus -1ex minus \n-.2ex}{2.3ex plus .2ex}{\\normalsize\\bf}}\n\\makeatother\n\n\n\n\\begin{document}\n\n\n\\title{\\textbf{\\large On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations}}\n\\author{\\textsc{\\normalsize Marjorie G.\\ Hahn, Kei Kobayashi, Jelena Ryvkina and Sabir Umarov}\\thanks{Department of Mathematics, \n Tufts University, 503 Boston Avenue, Medford, MA 02155, USA; marjorie.hahn@tufts.edu, kei.kobayashi@tufts.edu, jelena.ryvkina@tufts.edu, sabir.umarov@tufts.edu}\n \\vspace{1mm} \\\\ \\textit{\\small Tufts University}}\n\\date{ }\n\\maketitle \n\\vspace{-6mm}\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\begin{abstract}\nThis paper establishes Fokker-Planck-Kolmogorov type equations for time-changed\nGaussian processes. \nExamples include those equations for a time-changed fractional Brownian motion with time-dependent Hurst parameter and for a time-changed Ornstein-Uhlenbeck process. The time-change process considered is the inverse of either a stable subordinator or a mixture of independent stable subordinators.\\footnote[0]{\\textit{AMS 2010 subject classifications:} Primary 60G15, 35Q84; secondary 60G22. \n \\textit{Keywords:} time-change, inverse subordinator, Gaussian process, Fokker-Planck equation, Kolmogorov equation, fractional Brownian motion, time-dependent Hurst parameter, Volterra process.} \n\\end{abstract}\n\n\\section{Introduction} \n\\label{Sec1}\n\n\nA one-dimensional stochastic process $X=(X_t)_{t\\ge 0}$ is called a \\textit{Gaussian process} if the random vector $(X_{t_1},\\ldots,X_{t_m})$ has a multivariate Gaussian distribution for all finite sequences $0\\le t_1<\\cdots0,\n\\]\none can represent $D_{\\ast,t}^{\\beta}$ in the form\n$D_{\\ast,t}^{\\beta}=J^{1-\\beta}_t\\circ (d\/dt)$ (see \\cite{GM97} for details). \n\n\n\nFor the last few decades, time-fractional order FPKEs have appeared as an essential tool for the study of dynamics of various complex processes arising in anomalous diffusion in physics\n\\cite{MetzlerKlafter00,Zaslavsky}, finance \\cite{GMSR,Janczura}, hydrology\n\\cite{BWM} and cell biology \\cite{Saxton}.\nUsing several different methods, many authors derive FPKEs associated with time-changed stochastic processes.\nFor example, in \\cite{MMM-TAMS}, FPKEs for time-changed continuous Markov processes are obtained via the theory of semigroups. \nPaper \\cite{HKU} identifies a wide class of stochastic differential equations whose associated FPKEs are represented by time-fractional distributed order pseudo-differential equations. \nThe driving processes for these stochastic differential equations are time-changed L\\'evy processes. \nTwo different approaches are taken, one based on the semigroup technique and the other on the time-changed It\\^o formula in \\cite{Kobayashi}. \nPaper \\cite{HKU-2} provides FPKEs associated with a time-changed fractional Brownian motion from a functional-analytic viewpoint. \nContinuous time random walk-based approaches to derivations of time-fractional order FPKEs are illustrated in \\cite{GM1,MMM-ctrw,UG,US}. \nIn the current paper, we follow the method presented in \\cite{HKU-2}.\n\n\nConsider a time-change process $E^\\beta=(E^\\beta_t)_{t\\ge 0}$ given by the \\textit{inverse}, or the \\textit{first hitting time process}, of a $\\beta$-stable subordinator $W^\\beta=(W^\\beta_t)_{t\\ge 0}$ with $\\beta\\in(0,1)$. The relationship between the two processes is expressed as \n$E^\\beta_t=\\inf\\{s>0\\hspace{1pt};\\hspace{1pt} W^\\beta_s>t \\}$.\nTo make precise the problem being pursued in this paper, first recall the following results. For proofs of these results as well as analysis on the associated classes of stochastic differential equations, \nsee e.g.\\ \\cite{GM1,HKU,HKU-2,MMM-ctrw}.\nThroughout the paper, all processes are assumed to start at 0.\n\\begin{enumerate}\n \\item[(a)] If $E^\\beta$ is independent of an $n$-dimensional Brownian motion $B$, then $B$ and the time-changed Brownian motion $(B_{E^\\beta_t})$ have respective transition probabilities $p(t,x)$ and $q(t,x)$ satisfying the PDEs\n\\begin{align}\\label{Brownian FPK}\n \\partial_t \\hspace{1pt}p(t,x)=\\frac 12 \\varDelta \\hspace{1pt}p(t,x) \\ \\ \\ \\textrm{and} \\ \\ \\ D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)=\\frac 12 \\varDelta \\hspace{1pt}q(t,x), \n\\end{align}\nwhere $\\partial_t=\\frac{\\partial}{\\partial t}$ and $\\varDelta=\\sum_{j=1}^n \\partial_{x^j}^2=\\sum_{j=1}^n \\bigl(\\frac{\\partial}{\\partial {x^j}}\\bigr)^2$, with the vector $x\\in\\bbR^n$ denoted as $x=(x^1,\\ldots,x^n)$.\n \\item[(b)] Let $L$ be a L\\'evy process whose characteristic function is given by $\\bbE[e^{i(\\xi,L_t)}]=e^{t \\psi(\\xi)}$ with symbol $\\psi$ (see \\cite{Applebaum,Sato}). \nIf $E^\\beta$ is independent of $L$, then $L$ and the time-changed L\\'evy process $(L_{E^\\beta_t})$ have respective transition probabilities $p(t,x)$ and $q(t,x)$ satisfying the PDEs \n\\begin{align}\\label{Levy FPK}\n \\partial_t \\hspace{1pt}p(t,x)=\\mathcal{L}^\\ast p(t,x) \\ \\ \\ \\textrm{and} \\ \\ \\ \n D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)=\\mathcal{L}^\\ast q(t,x), \n\\end{align}\nwhere $\\mathcal{L}^\\ast$ is the conjugate of the pseudo-differential operator with symbol $\\psi$. \n \\item[(c)] If $E^\\beta$ is independent of an $n$-dimensional fractional Brownian motion $B^H$ with Hurst parameter $H\\in(0,1)$ (definition is given in Example \\hyperlink{Application-1}{1}), then $B^H$ and the time-changed fractional Brownian motion $(B^H_{E^\\beta_t})$ have respective transition probabilities $p(t,x)$ and $q(t,x)$ satisfying the PDEs\n\\begin{align}\\label{FBM FPK}\n \\partial_t \\hspace{1pt}p(t,x)=H t^{2H-1}\\varDelta \\hspace{1pt}p(t,x) \\ \\ \\ \\textrm{and} \\ \\ \\ \n D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)=H G^\\beta_{2H-1,t}\\varDelta \\hspace{1pt}q(t,x), \n\\end{align}\nwhere $G^\\beta_{\\gamma,t}$ with $\\gamma\\in(-1,1)$ is the operator acting on $t$ given by\n\\begin{equation} \\label{presentation}\nG^\\beta_{\\gamma,t} \\hspace{1pt}g(t)= \\beta \\Gamma(\\gamma + 1) J^{1-\\beta}_t\n\\mathcal{L}^{-1}_{s \\to t}\\biggl[\\frac{1}{2\\pi i} \\int_{C-i\n\\infty}^{C+i\\infty}\n\\frac{\\tilde{g}(z)}{(s^{\\beta}-z^{\\beta})^{\\gamma+1}}\\hspace{1pt}dz \\biggr](t),\n\\end{equation}\nwith $0 < C 0$ (see \\cite{Applebaum,Sato}). \nSince $W^\\beta$ is strictly increasing, its inverse $E^\\beta$ is\ncontinuous and nondecreasing, but no longer a L\\'evy process (see \\cite{MMM-infinitemean}). \nThe density $f_{E_t^\\beta}$ of $E_t^\\beta$ can be expressed using the density $f_{W_\\tau^\\beta}$ of $W_\\tau^\\beta$ as \n\\begin{align*} \nf_{E_t^\\beta}(\\tau) = \\partial_\\tau \\mathbb{P}(E^\\beta_t\\le \\tau)\n=\\partial_\\tau \\bigl\\{1-\\mathbb{P}(W^\\beta_\\tau < t)\\bigr\\}\n=-\\partial_\\tau \\bigl\\{[J^1_t f_{W_\\tau^\\beta}](t) \\bigr\\},\n\\end{align*}\nfrom which it follows that\n\\begin{align}\\label{laplace_Ebeta}\n \\mathcal{L}_{t \\to s}[f_{E_t^\\beta}(\\tau)](s)= -\\partial_\\tau \\biggl[\\frac{\\widetilde{f_{W_\\tau^\\beta}}(s)}{s}\\biggr]\n=s^{\\beta-1} e^{-\\tau s^\\beta}, ~ s>0, ~ \\tau \\ge 0.\n\\end{align}\nThe function $f_{E_t^\\beta}(\\tau)$ is $C^{\\infty}$ with respect to the two variables $t$ and $\\tau$.\n\n\n\nThe notion of time-change can be extended to the more general case where\nthe time-change process is given by the inverse of an arbitrary\nmixture of independent stable subordinators. \nLet $\\rho_\\mu(s)=\\int_0^1\ns^{\\beta}d \\mu (\\beta),$ where \n$\\mu$ is a finite measure with $\\textrm{supp}\\, \\mu \\subset (0,1).$\nLet $W^{\\mu}$ be a nonnegative stochastic\nprocess satisfying $\\mathbb{E}[ e^{-sW^{\\mu}_t}]=e^{-t \\rho_\\mu(s)}$ and let\n$E^{\\mu}_t=\\inf \\{\\tau>0\\hspace{1pt};\\hspace{1pt} W^{\\mu}_{\\tau} > t\\}.$\nClearly, $W^\\mu=W^{\\beta_0}$ if $\\mu=\\delta_{\\beta_0}$, the Dirac measure on $(0,1)$ concentrated on a single point $\\beta_0$. \nThe process\n$W^{\\mu}$ represents a weighted mixture of independent stable subordinators. \nSimilar to the identity \\eqref{laplace_Ebeta}, the density $f_{E_t^\\mu}$ of $E_t^\\mu$ has Laplace transform\n\\begin{align}\\label{laplace_Emu}\n \\mathcal{L}_{t \\to s}[f_{E_t^\\mu}(\\tau)](s)\n=\\frac{\\rho_\\mu(s)}{s} e^{-\\tau \\rho_\\mu(s)}, ~ s>0, ~ \\tau \\ge 0.\n\\end{align}\nFor further properties of $f_{E_t^\\beta}(\\tau)$ and $f_{E_t^\\mu}(\\tau)$, see \\cite{HKU-2,MMM-ctrw}.\n\n\n\nThe above time-change process $E^\\mu$ is connected with the \\textit{fractional derivative $D^\\mu_{\\ast}$ with distributed orders} given by \n\\begin{align}\\label{distributed-derivative}\n D^\\mu_{\\ast}\\hspace{1pt}g(t)=D^\\mu_{\\ast,t}\\hspace{1pt}g(t)=\\int_0^1 D^\\beta_\\ast g(t)\\hspace{1pt} d\\mu(\\beta). \n\\end{align}\nNamely, if $E^\\beta$ is replaced by $E^\\mu$ in items (a) and (b) of the list of known results in Section \\ref{Sec1}, then the FPKEs for the time-changed Brownian motion $(B_{E^\\mu_t})$ and the time-changed L\\'evy process $(L_{E^\\mu_t})$ are respectively given by (see \\cite{HKU})\n\\[\n D^\\mu_{\\ast,t} \\hspace{1pt}q(t,x)=\\frac 12 \\varDelta\\hspace{1pt} q(t,x)\n \\ \\ \\ \\textrm{and} \\ \\ \\ \n D^\\mu_{\\ast,t} \\hspace{1pt}q(t,x)=\\mL^\\ast q(t,x). \n\\] \nFor investigations into PDEs with fractional derivatives with distributed orders, see \\cite{Kochubey,MMM-ultraslow,UG05}.\n\n\n\nThe covariance function of a given zero-mean Gaussian process is symmetric and positive semi-definite; conversely, every symmetric, positive semi-definite function on $[0,\\infty)\\times[0,\\infty)$ is the covariance function of some zero-mean Gaussian process (see e.g., Theorem 8.2 of \\cite{Janson}). Examples of such functions include $R_X(s,t)=s \\wedge t$ for Brownian motion and $R_X(s,t)=\\sigma_0^2+s\\cdot t$ which is obtained from linear regression\n (see \\cite{RW}). \nThe sum and the product of two covariance functions for Gaussian processes are again covariance functions for some Gaussian processes. For more examples of covariance functions, consult e.g.\\ \\cite{RW}. \n\n\n\nAn $n$-dimensional Gaussian process $X=(X^1,\\ldots,X^n)$ is a process whose components $X^j$ are independent one-dimensional Gaussian processes with possibly distinct covariance functions $R_{X^j}(s,t)$.\nThe variance functions $R_{X^j}(t)=R_{X^j}(t,t)$ will play\nan important role in establishing FPKEs for Gaussian and time-changed Gaussian processes. \nFor differentiable variance functions $R_{X^j}(t)$, let\n\\begin{align}\\label{FP-volterra0.5}\n A=A_X=\\frac 12\\sum_{j=1}^n R_{X^j}'(t)\\hspace{1pt}\\partial_{x^j}^2.\n\\end{align}\nClearly $A=\\frac{1}{2}\\hspace{1pt}\\varDelta$ if $X$ is an $n$-dimensional Brownian motion.\n\n\n\n\n\n\\begin{proposition} \\label{proposition_FP-volterra}\nLet $X=(X^1,\\ldots,X^n)$ be an $n$-dimensional zero-mean Gaussian process with covariance functions $R_{X^j}(s,t)$, $j=1,\\ldots,n$. Suppose the variance functions $R_{X^j}(t)=R_{X^j}(t,t)$ are differentiable on $(0,\\infty)$. \nThen the transition probabilities $p(t,x)$ of $X$ satisfy the PDE\n\\begin{align} \\label{FP-volterra1}\n \\partial_t \\hspace{1pt}p(t,x)= A\\hspace{1pt} p(t,x), \\ t>0, \\ x\\in\\bbR^n,\n\\end{align}\nwhere $A$ is the operator in \\eqref{FP-volterra0.5}.\n\\end{proposition}\n\n\\begin{proof}\nSince the components $X^j$ of $X$ are independent zero-mean Gaussian processes, it follows that\n\\begin{align}\np(t,x)=\\prod_{j=1}^n\\bigl(2\\pi R_{X^j}(t)\\bigr)^{-1\/2}\\times \\exp\\biggl\\{-\\sum_{j=1}^n\\frac{(x^j)^2}{2R_{X^j}(t)}\\biggr\\}.\n\\end{align}\nDirect computation of partial derivatives of $p(t,x)$\nyields the equality in \\eqref{FP-volterra1}.\n\\end{proof}\n\n\n\\begin{remark}\n\\begin{em}\na) In Proposition \\ref{proposition_FP-volterra}, if the components $X^j$ are independent Gaussian processes with a \\textit{common} variance function $R_{X}(t)$ which is differentiable, then \n\\eqref{FP-volterra1} reduces to the following form which agrees with the classical FPKE:\n\\begin{align} \n \\partial_t \\hspace{1pt}p(t,x)=\\frac 12 R_X'(t) \\varDelta \\hspace{1pt}p(t,x).\n\\end{align}\n\nb) If $X=(X^1,\\ldots,X^n)$ is an $n$-dimensional Gaussian process with mean functions $m_{X^j}(t)$ and covariance functions $R_{X^j}(s,t)$, and if both $m_{X^j}(t)$ and $R_{X^j}(t)=R_{X^j}(t,t)$ are differentiable, \nthen the associated FPKE contains an additional term:\n\\begin{align} \n \\partial_t \\hspace{1pt}p(t,x)=A \\hspace{1pt}p(t,x)\n + B \\hspace{1pt}p(t,x) \\ \\ \\ \\ \\textrm{where} \\ \\ \\ \\ \n B=-\\sum_{j=1}^n m_{X^j}'(t)\\hspace{1pt} \\partial_{x^j}.\n\\end{align}\nSuch Gaussian processes include e.g.\\ the process defined by the sum of a Brownian motion and a deterministic differentiable function.\n\nc) The initial distribution of the Gaussian process $X$ in Proposition \\ref{proposition_FP-volterra} is not given; however, it needs to be specified in order to guarantee uniqueness of the solution to PDE \\eqref{FP-volterra1}. The same argument applies to the PDEs to be established in Theorems \\ref{theorem_FP-volterra} and \\ref{theorem_FP-volterra_general} as well.\n\\end{em}\n\\end{remark}\n\n\n\n\n\n\\section{FPKEs for time-changed Gaussian processes}\n\\label{Sec3}\n\n\nTheorems \\ref{theorem_FP-volterra} and \\ref{theorem_FP-volterra_general} formulate the FPKE for a time-changed Gaussian process under the assumption that the time-change process is independent of the Gaussian process. \nThe case where those processes are dependent is not discussed in this paper. \nAs in Proposition \\ref{proposition_FP-volterra}, the variance function plays a key role here. \n\n\n\\begin{theorem} \\label{theorem_FP-volterra} \nLet $X=(X^1,\\ldots,X^n)$ be an $n$-dimensional \nzero-mean Gaussian process with covariance functions $R_{X^j}(s,t)$, $j=1,\\ldots,n$,\nand let $E^\\beta$ be the inverse of a stable subordinator $W^\\beta$ of index $\\beta\\in(0,1)$, independent of $X$. \n Suppose the variance functions $R_{X^j}(t)=R_{X^j}(t,t)$ are differentiable on $(0,\\infty)$ and Laplace transformable. \nThen the transition probabilities $q(t,x)$ of the time-changed Gaussian process $(X_{E^\\beta_t})$ satisfy the equivalent PDEs\n\\begin{align} \\label{FP-volterra3-2}\n {D}^\\beta_{\\ast,t} \\hspace{1pt}q(t,x)&=\n\\sum_{j=1}^n J^{1-\\beta}_t \\Lambda^\\beta_{X^j,t}\\hspace{1pt} \\partial_{x^j}^2 \\hspace{1pt}q(t,x), \\ t>0, \\ x\\in\\bbR^n,\\\\\n\\label{FP-volterra3-1}\n \\partial_t \\hspace{1pt}q(t,x)&= \n\\sum_{j=1}^n \\Lambda^\\beta_{X^j,t}\\hspace{1pt} \\partial_{x^j}^2 \\hspace{1pt}q(t,x), \\ t>0, \\ x\\in\\bbR^n,\n\\end{align}\nwhere $\\Lambda^\\beta_{X^j,t}$, $j=1,\\ldots,n$, are the operators acting on $t$ given by\n\\begin{align} \\label{FP-volterra3.5}\n \\Lambda^\\beta_{X^j,t}\\hspace{1pt} g(t)=\n \\frac\\beta 2\\hspace{1pt} \\mL^{-1}_{s\\to t} \n \\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}}\n (s^\\beta-z^\\beta)\\widetilde{R_{X^j}}(s^\\beta-z^\\beta) \n \\hspace{1pt}\\tilde{g}(z)\\hspace{1pt} dz\\biggr] (t),\n\\end{align}\nwith $z^\\beta=e^{\\beta \\hspace{1pt}\\textrm{\\rm Ln}(z)}$, $\\textrm{\\rm Ln}(z)$ being the principal value of the complex logarithmic function $\\ln(z)$ \nwith cut along the negative real axis,\nand \n$\\bm{\\mathcal{C}}$ being a curve in the complex plane obtained via the transformation $\\zeta=z^\\beta$ which leaves all the singularities of $\\widetilde{R_{X^j}}$ on one side.\n\\end{theorem}\n\n\n\\begin{proof}\nLet $p(t,x)$ denote the transition probabilities of the Gaussian process $X$. \nFor each $x\\in\\bbR^n$, it follows from the independence assumption between $E^\\beta$ and $X$ that \n\\begin{equation} \\label{relation}\n q(t,x)=\\int_0^{\\infty} f_{E_t^\\beta}(\\tau)\\hspace{1pt}p(\\tau,x)\\;d\\tau, \\ t>0.\n\\end{equation}\nRelationship \\eqref{relation} and equality \\eqref{laplace_Ebeta} together yield \n\\begin{align} \\label{FP-volterra4}\n \\tilde{q}(s,x)=s^{\\beta-1}\\tilde{p}(s^\\beta,x), \\ s>0.\n\\end{align}\nSince $R_{X^j}(t)$ is Laplace transformable, \n$\\widetilde{R_{X^j}}(s)$ exists for all $s>a$, for some constant $a\\ge 0$. \nTaking Laplace transforms on both sides of \\eqref{FP-volterra1}, \n\\begin{align} \\label{FP-volterra4.5}\n s \\,\\tilde{p}(s,x)-p(0,x)\n &=\\frac 12\\sum_{j=1}^n\\mL_{t\\to s}[ R_{X^j}'(t) \\partial_{x^j}^2 \\hspace{1pt} p(t,x)](s)\\\\\n &=\\frac 12 \\sum_{j=1}^n \\bigl(\\mL_{t}[R_{X^j}'(t)]\\ast \\mL_{t}[\\partial_{x^j}^2 \\hspace{1pt} p(t,x)] \\bigr)(s)\\notag \\\\\n &=\\frac 12 \\sum_{j=1}^n \\biggl[ \\frac 1{2\\pi i} \\int_{c-i\\infty}^{c+i\\infty} \\widetilde{R_{X^j}'}(s-\\zeta) \\partial_{x^j}^2 \\hspace{1pt} \\tilde{p}(\\zeta,x) \\hspace{1pt}d\\zeta\\biggr]\\notag\\\\\n &=\\frac \\beta 2 \\sum_{j=1}^n\\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}} \\widetilde{R_{X^j}'}(s-z^\\beta) \\partial_{x^j}^2 \\hspace{1pt} \\tilde{p}(z^\\beta,x) z^{\\beta-1}\\hspace{1pt}dz\\biggr], \\ s>a, \\notag\n\\end{align}\nwhere $\\ast$ denotes the convolution of Laplace images and \nthe function \n\\begin{align} \\label{FP-volterra4.6}\n \\widetilde{R_{X^j}'}(s)=s\\hspace{1pt}\\widetilde{R_{X^j}}(s), \\ s>a,\n\\end{align}\nexists by assumption. \nEquation \\eqref{FP-volterra4.6} is valid since $R_{X^j}(0)=0$ due to the initial condition $X^j(0)=0$.\nSince $E^\\beta_0=0$ with probability one, it follows that $p(0,x)=q(0,x)$. Replacing $s$ by $s^\\beta$ and using the identity \\eqref{FP-volterra4} yields\n\\begin{align}\\label{FP-volterra5}\n s \\,\\tilde{q}(s,x)-q(0,x)\n =\\frac \\beta 2 \\sum_{j=1}^n \\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}} \\widetilde{R_{X^j}'}(s^\\beta-z^\\beta) \\partial_{x^j}^2 \\hspace{1pt}\\tilde{q}(z,x) \\hspace{1pt}dz\\biggr], \\ s>a^{1\/\\beta}.\n\\end{align}\nSince the left hand side equals $\\mL_{t\\to s}[\\partial_t \\hspace{1pt}q(t,x)](s)$, PDE \\eqref{FP-volterra3-1} follows upon\nsubstituting \\eqref{FP-volterra4.6}\nand taking the inverse Laplace transform on both sides.\nMoreover, applying the fractional integral operator $J^{1-\\beta}_t$ to both sides of \\eqref{FP-volterra3-1} yields \\eqref{FP-volterra3-2}.\n\\end{proof}\n\nThe next theorem extends the previous theorem to time-changes which are the inverses of mixtures of independent stable subordinators. \n\n\n\\begin{theorem} \\label{theorem_FP-volterra_general} \nLet $X=(X^1,\\ldots,X^n)$ be an $n$-dimensional \nzero-mean Gaussian process with covariance functions $R_{X^j}(s,t)$, $j=1,\\ldots,n$, and let\n$E^\\mu$ be the inverse of a mixture $W^\\mu$ of independent stable subordinators, independent of $X$. \nSuppose the variance functions $R_{X^j}(t)=R_{X^j}(t,t)$ are differentiable on $(0,\\infty)$ and Laplace transformable. \nThen the transition probabilities $q(t,x)$ of the time-changed Gaussian process $(X_{E^\\mu_t})$ satisfy the PDEs\n\\begin{align} \\label{FP-volterra13-2}\n D^\\mu_{\\ast,t} \\hspace{1pt}q(t,x)= \n\\sum_{j=1}^n \n\\int_0^1 J^{1-\\beta}_t \\Lambda^\\mu_{X^j,t}\\hspace{1pt} \\partial_{x^j}^2 \\hspace{1pt} q(t,x) \\hspace{1pt} d\\mu(\\beta), \\ t>0, \\ x\\in\\bbR^n,\n\\end{align}\nand\n\\begin{align} \\label{FP-volterra13-1}\n \\partial_t \\hspace{1pt}q(t,x)=\n\\sum_{j=1}^n \\Lambda^\\mu_{X^j,t}\\hspace{1pt} \\partial_{x^j}^2 \\hspace{1pt}q(t,x), \\ t>0, \\ x\\in\\bbR^n,\n\\end{align}\nwhere $D^\\mu_{\\ast,t}$ is the operator in \\eqref{distributed-derivative} and $\\Lambda^\\mu_{X^j,t}$, $j=1,\\ldots,n$, are the operators acting on $t$ given by\n\\begin{align}\n \\Lambda^\\mu_{X^j,t}\\hspace{1pt} g(t)\n =\\frac {1}{2} \\hspace{1pt} \\mL^{-1}_{s\\to t} \n \\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}}\n \\bigl(\\rho_\\mu(s)-\\rho_\\mu(z)\\bigr) \\widetilde{R_{X^j}}\\bigl(\\rho_\\mu(s)-\\rho_\\mu(z)\\bigr) \n \\hspace{1pt}m_\\mu(z) \\hspace{1pt}\\tilde{g}(z)\\hspace{1pt}dz\\biggr] (t),\n\\end{align}\nwith $\\rho_\\mu(z)=\\int_0^1 e^{\\beta \\hspace{1pt} \\emph{Ln} (z)}\\hspace{1pt}d\\mu(\\beta)$, $m_{\\mu}(z)=\\frac{1}{\\rho_\\mu(z)}\\int_0^1 \\beta z^{\\beta}\\hspace{1pt} d \\mu(\\beta)$, and \n$\\bm{\\mathcal{C}}$ being a curve in the complex plane obtained via the transformation $\\zeta=\\rho_\\mu(z)$ which leaves all the singularities of $\\widetilde{R_{X^j}}$ on one side.\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe only sketch the proof since it is similar to the proof of Theorem\n\\ref{theorem_FP-volterra}. \nLet $p(t,x)$ denote the transition probabilities of the Gaussian process $X$. For each $x\\in\\bbR^n$, it follows from relationship \\eqref{relation} with $f_{E^\\beta_t}$ replaced by $f_{E^\\mu_t}$ together with equality \\eqref{laplace_Emu} that\n\\begin{align} \\label{FP-volterra14}\n \\tilde{q}(s,x)=\\frac{\\rho_\\mu(s)}{s}\\hspace{1pt}\\tilde{p}\\bigl(\\rho_\\mu(s),x\\bigr), \\ s>0.\n\\end{align}\nTaking Laplace transforms on both sides of \\eqref{FP-volterra1} leads to the second to last equality in \\eqref{FP-volterra4.5}. Letting $\\zeta=\\rho_{\\mu}(z)$ yields\n\\begin{align*} \n s \\,\\tilde{p}(s,x)-p(0,x)=\\frac 1 2 \\sum_{j=1}^n \\biggl[ \\frac 1{2\\pi i} \\int_{\\bm{\\mathcal{C}}} \\widetilde{R_{X^j}'}\\bigl(s-\\rho_\\mu(z)\\bigr) \\partial_{x^j}^2 \\hspace{1pt} \\tilde{p}\\bigl(\\rho_\\mu(z),x\\bigr) \\frac {\\rho_\\mu(z)}{z}\\hspace{1pt} m_\\mu(z) \\hspace{1pt}dz\\biggr],\n\\end{align*}\nwhich is valid for all $s$ for which $\\widetilde{R_{X^j}}(s)$ exists.\nReplacing $s$ by $\\rho_\\mu(s)$ and using the identity \\eqref{FP-volterra14} yields an equation similar to \\eqref{FP-volterra5}. \nPDE \\eqref{FP-volterra13-1} is obtained upon taking the inverse Laplace transform on both sides. Finally, applying the fractional integral operator $J^{1-\\beta}_t$ and integrating with respect to $\\mu$ on both sides of \\eqref{FP-volterra13-1} yields \\eqref{FP-volterra13-2}.\n\\end{proof}\n\n\n\\begin{remark}\n\\begin{em}\na) If $\\mu=\\delta_{\\beta_0}$ with $\\beta_0\\in(0,1)$, then $\\Lambda^\\mu_{X^j,t} \\hspace{1pt}g(t)=\\Lambda^{\\beta_0}_{X^j,t}\\hspace{1pt}g(t)$ and the FPKEs in \\eqref{FP-volterra13-2} and \\eqref{FP-volterra13-1}\nrespectively reduce to the FPKEs in \\eqref{FP-volterra3-2} and \\eqref{FP-volterra3-1} with $\\beta=\\beta_0$, as expected. \n\nb) In Theorem \\ref{theorem_FP-volterra_general}, if the components $X^j$ are independent Gaussian processes with a \\textit{common} variance function $R_{X}(t)$ which is differentiable and Laplace transformable, then the FPKEs in \\eqref{FP-volterra13-2} and \\eqref{FP-volterra13-1} respectively reduce to the following simple forms:\n\\begin{align} \n D^\\mu_{\\ast,t} \\hspace{1pt}q(t,x) &= \\hspace{1pt} \\int_0^1 J^{1-\\beta}_t \\Lambda^\\mu_{X,t} \\hspace{1pt}\\varDelta \\hspace{1pt} q(t,x)\\hspace{1pt} d\\mu(\\beta);\\\\\n \\partial_t \\hspace{1pt}q(t,x)&= \\Lambda^\\mu_{X,t} \\varDelta \\hspace{1pt} q(t,x).\n\\end{align}\n\nc) As Example \\hyperlink{Application-1}{1} in Section \\ref{Sec4} shows, Theorem \\ref{theorem_FP-volterra} extends the time-fractional order FPKE in \\eqref{FBM FPK} for a time-changed fractional Brownian motion to that for a general time-changed Gaussian process, revealing the role of the variance function in describing the dynamics of the process.\n\\end{em}\n\\end{remark}\n\n\n\\section{Applications} \\label{Sec4}\n\n\n\nThis section is devoted to applications of the results established in this paper concerning FPKEs for Gaussian and time-changed Gaussian processes. For simplicity of discussion, we will consider the time-changed process $E^\\beta$ given by the inverse of a stable subordinator $W^\\beta$ of index $\\beta\\in(0,1)$, rather than the more general time-change process $E^\\mu$.\n\n\\vspace{3.2mm}\n\n\\noindent\n\\textsl{Example \\hypertarget{Application-1}{1}. Fractional Brownian motion.}\nOne of the most important Gaussian processes in applied probability is a fractional Brownian motion $B^H$ with Hurst parameter $H\\in(0,1)$. \nA one-dimensional \\textit{fractional Brownian motion} is a zero-mean Gaussian process with covariance function\n\\begin{equation}\\label{FBM-cov}\nR_{B^H}(s,t)=\\bbE[B_s^H B_t^H]=\\frac{1}{2} (s^{2H}+t^{2H}- |s-t|^{2H}).\n\\end{equation}\nIf $H=1\/2,$ then \n$B^H$ becomes a usual Brownian motion. \nAn $n$-dimensional fractional Brownian motion is\nan $n$-dimensional process whose components are independent fractional Brownian motions with a common Hurst parameter.\n\nStochastic processes driven by a fractional Brownian motion $B^H$ are of increasing interest for both theorists and applied researchers due\nto their wide application in fields such as mathematical finance\n\\cite{Cheridito03}, solar activities \\cite{solar} and\nturbulence \\cite{turbulence}.\nThe process $B^H,$ like the usual Brownian motion,\nhas nowhere differentiable paths and stationary increments; however, it does not have independent increments. \n$B^H$ has the integral representation \n$B_t^H=\\int_0^t K_H(t,s)\\hspace{1pt}d B_s,$ where $B$ is a Brownian motion and $K_H(t,s)$ is a deterministic kernel. \n$B^H$ is not a semimartingale unless $H=1\/2$, so the usual It\\^o's stochastic calculus is not valid. \nFor details of the above properties, see \\cite{book,Nualart}. \n\n\n\nLet $B^H$ be an $n$-dimensional fractional Brownian motion and let $E^\\beta$ be the inverse of a stable subordinator of index $\\beta\\in(0,1)$, independent of $B^H$. Then the components of $B^H$ share the common variance function $R_{B^H}(t)=t^{2H}$ and its Laplace transform $\\widetilde{R_{B^H}'}(s)=2H\\hspace{1pt} \\Gamma(2H)\/s^{2H}$. \nHence, Proposition \\ref{proposition_FP-volterra} and Theorem \\ref{theorem_FP-volterra} immediately recover both the FPKEs in \\eqref{FBM FPK} for the fractional Brownian motion $B^H$ and the time-changed fractional Brownian motion $(B^H_{E^\\beta_t})$. \nIn this case,\n\\begin{align} \\label{correspondence-operators}\nJ^{1-\\beta}_t \\Lambda^\\beta_{B^H,t}=H G^\\beta_{2H-1,t},\n\\end{align}\nwhere $G^\\beta_{\\gamma,t}$ is the operator given in \\eqref{presentation}. \nNote that the curve $\\bm{\\mathcal{C}}$ appearing in the expression of the operator $\\Lambda^\\beta_{B^H,t}$ in \\eqref{FP-volterra3.5} can be replaced by a vertical line $\\{C+i r \\hspace{1pt}; \\hspace{1pt}r\\in \\bbR\\}$ with $0s, \n\\end{align}\nwhere the positive constant $c_H$ is chosen so that the integral\n$\\int_0^{t\\wedge s} K_H(t,r)K_H(s,r) \\hspace{1pt}dr$ coincides with \n$R_{B^H}(s,t)$ in \\eqref{FBM-cov}.\nIncrements of $B^H$ exhibit long range dependence.\n\n\nA particular interesting Volterra process\nis the fractional Brownian motion with time-dependent Hurst parameter $H(t)$ suggested in Theorem 9 of \\cite{Decr-volterra}.\nNamely, suppose $H(t): [0,T]\\rightarrow (1\/2,1)$ is a deterministic function satisfying the following conditions:\n\\begin{align}\\label{condition_star}\n \\inf_{t\\in[0,T]} H(t)>\\frac 12 \\ \\hspace{3pt} \\textrm{and} \\ \\hspace{3pt} \n H(t)\\in \\mathcal{S}_{1\/2+\\alpha,2} \\ \\hspace{3pt}\\textrm{for some} \\ \\hspace{3pt} \n \\alpha\\in\\biggl(0,\\hspace{2pt}\\inf_{t\\in[0,T]}H(t)-\\frac 12\\biggr), \n\\end{align}\nwhere $\\mathcal{S}_{\\eta,2}$ is the Sobolev-Slobodetzki space given by the closure of the space $C^1[0,T]$ with respect to the semi-norm \n\\begin{align}\n \\| f \\|^2= \\int_0^T\\hspace{-1.2mm} \\int_0^T \\frac{|f(t)-f(s)|^2}{|t-s|^{1+2\\eta}}\\hspace{1pt} dt \\hspace{1pt}ds. \n\\end{align}\nThen representation \\eqref{K_H-1} with $H$ replaced by $H(t)$ induces a covariance function $R_{V}(s,t)=\\int_0^{t\\wedge s} K_{H(t)}(t,r) \\hspace{1pt}K_{H(s)}(s,r)\\hspace{1pt}dr$ for some Volterra process $V$ on $[0,T]$. \nThe variance function is given by $R_{V}(t)=t^{2H(t)}$ and is necessarily continuous due to the Sobolev embedding theorem, which says $\\mathcal{S}_{\\eta,2}\\subset C[0,T]$ for all $\\eta>1\/2$ (see e.g.\\ \\cite{Holmander3}).\nTherefore, $H(t)$ is also continuous. \n\n\nLet $H(t): [0,\\infty) \\rightarrow (1\/2,1)$ be a differentiable function\nwhose restriction to any finite interval $[0,T]$ satisfies the conditions in \\eqref{condition_star}. \nFor each $T>0$, let $K_{V^T}(s,t)$ be the kernel inducing the covariance function $R_{V^T}(s,t)$ of the associated Volterra process $V^T$ defined on $[0,T]$ as above. \nThe definition of $K_{V^T}(s,t)$ is consistent; i.e.\\ $K_{V^{T_1}}(s,t)=K_{V^{T_2}}(s,t)$ for any $0\\le s,\\hspace{1pt} t\\le T_1\\le T_2<\\infty$. \nHence, so is that of $R_{V^T}(s,t)$, which implies that the function $R_X(s,t)$ given by $R_X(s,t)=R_{V^T}(s,t)$ whenever $0\\le s,\\hspace{1pt} t\\le T<\\infty$ is \na well-defined covariance function of a Gaussian process $X$ on $[0,\\infty)$ whose restriction to each interval $[0,T]$ coincides with $V^T$.\nThe process $X$ represents a fractional Brownian motion with variable Hurst parameter. \nThe variance function $R_X(t)=t^{2H(t)}$ is differentiable on $(0,\\infty)$ by assumption and Laplace transformable due to the estimate $R_X(t)\\le t^2$. Therefore, Proposition \\ref{proposition_FP-volterra} and Theorem \\ref{theorem_FP-volterra} can be applied to yield the FPKEs for $X$ and the time-changed process $(X_{E^\\beta_t})$ under the independence assumption between $E^\\beta$ and $X$.\n\n\n\n\n\n\n\\vspace{3.2mm}\n\n\\noindent\n\\textsl{Example \\hypertarget{Application-4}{4}. Fractional Brownian motion with piecewise constant Hurst parameter.} \nThe fractional Brownian motion discussed in Example \\hyperlink{Application-3}{3} has a continuously varying Hurst parameter $H(t): [0,\\infty) \\rightarrow (1\/2,1)$. Here we consider a piecewise constant Hurst parameter $H(t): [0,\\infty) \\rightarrow (0,1)$ which is described as\n\\begin{align}\\label{H(t)}\nH(t)=\\sum_{k=0}^N H_k \\textbf{\\textit{I}}_{[T_k,T_{k+1})}(t),\n\\end{align} \nwhere $\\{H_k\\}_{k=0}^N$ are constants in $(0,1)$, $\\{T_k\\}_{k=0}^N$ are fixed times such that $0=T_00$, $y_0\\in\\bbR$ are constants and $B$ is a standard Brownian motion. \nIf $\\alpha=0$, then $Y_t=y_0+\\sigma B_t$, a Brownian motion multiplied by $\\sigma$ starting at $y_0$.\nSuppose $\\alpha>0$. The process $Y$ defined by \\eqref{example_OU-1} is the unique strong solution to the inhomogeneous linear SDE\n \\begin{align}\n dY_t=-\\alpha Y_tdt +\\sigma dB_t \\ \\ \\textrm{with} \\ \\ Y_0=y_0, \\label{SDE81-1}\n \\end{align}\nwhich is associated with the SDE \n \\begin{align}\n d\\bar{Y}_t=-\\alpha \\bar{Y}_tdE^\\beta_t +\\sigma dB_{E^\\beta_t} \\ \\ \\textrm{with} \\ \\ \\bar{Y}_0=y_0, \\label{SDE81-2}\n \\end{align}\nvia the dual relationships $\\bar{Y}_t=Y_{E^\\beta_t}$ and $Y_t=\\bar{Y}_{W^\\beta_t}$; \nsee \\cite{Kobayashi} for details.\n\nConsider the zero-mean process $X$ defined by\n\\begin{align}\n X_t=Y_t-y_0\\hspace{1pt}e^{-\\alpha t}=\\sigma \\int_0^t e^{-\\alpha(t-s)}\\hspace{1pt}dB_s.\n\\end{align} \n$X$ is a Gaussian process since each random variable $X_t$ is a linear transformation of the It\\^o stochastic integral of the deterministic integrand $e^{\\alpha s}$.\nDirect calculation yields \n$R_X(t)=\\frac{\\sigma^2}{2\\alpha}(1-e^{-2\\alpha t})$ and\n$\\widetilde{R'_{X}}(s)=\\frac{\\sigma^2}{s+2\\alpha}$. \nTherefore, due to Proposition \\ref{proposition_FP-volterra} and Theorem \\ref{theorem_FP-volterra}, the initial value problems associated with $X$ and $(X_{E^\\beta_t})$, where $E^\\beta$ is independent of $X$, are respectively given by \n\\begin{align}\\label{example_OU-2}\n \\partial_t \\hspace{1pt}p(t,x)&=\\frac{\\sigma^2}{2}\\hspace{1pt} e^{-2\\alpha t}\\hspace{1pt}\\partial_x^2 \\hspace{1pt}p(t,x), \\ p(0,x)=\\delta_{0}(x);\\\\\n\\label{example_OU-3} D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)&=\\frac{\\sigma^2 \\beta}{2} \\hspace{1pt}J^{1-\\beta}_t\n\\mathcal{L}^{-1}_{s \\to t}\\biggl[\\frac{1}{2\\pi i} \\int_{\\bm{\\mathcal{C}}}\n\\frac{\\partial_x^2\\hspace{1pt}\\tilde{q}(z,x)}{s^\\beta-z^\\beta+2\\alpha}\\; dz \\biggr](t), \\ q(0,x)=\\delta_{0}(x).\n\\end{align}\nThe unique representation of the solution to the initial value problem \\eqref{example_OU-2} is obtained via the usual technique using the Fourier transform. Moreover, expression \\eqref{relation} guarantees uniqueness of the solution to \\eqref{example_OU-3} as well.\n\n\n\nNotice that the two processes $X$ and $(X_{E^\\beta_t})$ are unique strong solutions to SDEs \\eqref{SDE81-1} and \\eqref{SDE81-2} with $y_0=0$, respectively. Therefore, it is also possible to apply Theorem 4.1 of \\cite{HKU} to obtain the following forms of initial value problems which are understood in the sense of generalized functions: \n\\begin{align}\\label{example_OU-4}\n \\partial_t \\hspace{1pt}p(t,x)&=\\alpha \\hspace{1pt}\\partial_x \\bigl\\{x p(t,x) \\bigr\\} + \\frac{\\sigma^2}{2}\\hspace{1pt} \\partial_x^2 \\hspace{1pt}p(t,x), \\ p(0,x)=\\delta_{0}(x);\\\\\n\\label{example_OU-5} D_{\\ast,t}^{\\beta}\\hspace{1pt}q(t,x)&=\\alpha \\hspace{1pt}\\partial_x \\bigl\\{x q(t,x) \\bigr\\} + \\frac{\\sigma^2}{2}\\hspace{1pt} \\partial_x^2 \\hspace{1pt}q(t,x), \\ q(0,x)=\\delta_{0}(x).\n\\end{align}\nActually these FPKEs hold in the strong sense as well. \nFor uniqueness of solutions to \\eqref{example_OU-4} and \\eqref{example_OU-5}, see e.g.\\ \\cite{Friedman} and Corollary 3.2 of \\cite{HKU}.\n\n\nThe above discussion yields the following two sets of equivalent initial value problems: \\eqref{example_OU-2} and \\eqref{example_OU-4}, and \\eqref{example_OU-3} and \\eqref{example_OU-5}.\nAt first glance, PDE \\eqref{example_OU-2} might seem simpler or computationally more tractable than PDE \\eqref{example_OU-4}; however, PDE \\eqref{example_OU-3} which is associated with the time-changed process has a more complicated form than PDE \\eqref{example_OU-5}. \nA significant difference between PDEs \\eqref{example_OU-2} and \\eqref{example_OU-4} is the fact that the right-hand side of \\eqref{example_OU-4} can be expressed as $A^\\ast p(t,x)$ with the \\textit{spatial} operator $A^\\ast=\\alpha \\hspace{1pt}\\partial_x x + \\frac{\\sigma^2}{2}\\hspace{1pt} \\partial_x^2$ whereas the right-hand side of \\eqref{example_OU-2} involves both the spatial operator $\\partial_x^2$ and the \\textit{time-dependent} multiplication operator by $e^{-2\\alpha t}$. This observation suggests: 1) establishing FPKEs for time-changed processes via several different forms of FPKEs for the corresponding untime-changed processes, and 2) choosing appropriate forms for handling specific problems. \n\n\n\n\n\\begin{remark}\n\\begin{em}\nExample \\hyperlink{Application-5}{5} treated a Gaussian process given by the It\\^o integral of the deterministic integrand $e^{-\\alpha(t-s)}$. \nMalliavin calculus, which is valid for an arbitrary Gaussian integrator, can be regarded as an extension of It\\^o integration (see \\cite{Janson,Malliavin,Nualart}). It is known that Malliavin-type stochastic integrals of deterministic integrands are again Gaussian processes. Therefore, if the variance function of such a stochastic integral satisfies the technical conditions specified in Theorem \\ref{theorem_FP-volterra}, then the FPKE for the time-changed stochastic integral is explicitly given by \n\\eqref{FP-volterra3-2}, or equivalently, \\eqref{FP-volterra3-1}. \n\\end{em}\n\\end{remark}\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOne of the most important observable predictions for star formation theory is the resulting spectrum of masses, known as the initial mass function (IMF). The extreme ends of this distribution are of particular interest for constraining star formation physics. At the lowest masses, brown dwarfs (BDs) are thought to be approximately as numerous in the Milky Way as hydrogen-burning stars \\citep{Cha03}. However, what determines the IMF, and whether it varies with environment, remain the topic of debate \\citep[see][for a review]{Kru14}. \n\n\nBDs are usually defined to be objects less massive than required to burn ordinary hydrogen ($\\lesssim 0.08\\, M_\\odot$) and greater than the deuterium-burning limit ($\\gtrsim 13\\, M_\\mathrm{J}$). If the IMF in this regime is set by the local thermal Jeans mass, then the lower mass limit for fragmentation is set by the requirement that the centre of the clump can cool \\citep[i.e. the opacity limit --][]{Low76}. In this case, as the metallicity increases, the cooling becomes more efficient and the minimum mass for BDs decreases \\citep{Bat05}. However, dynamical processes and competitive accretion might further influence the distribution of stellar masses \\citep[e.g.][]{Bon97, Kle98,Bon08, Bat12}. Dynamical interactions can eject stars from their formation environment, shutting off accretion and stunting growth before the hydrogen mass burning limit is reached \\citep{Rei01, Bat02b}. Alternatively, the photoevaporation of the gas in an accreting envelope due to irradiation by neighbouring OB stars can have a similar effect \\citep{Hes96, Whi04}.\n\nThe low-mass IMF appears to have a similar shape across numerous local star forming regions \\citep{Andersen08}. However, a number of regions also appear to exhibit deviations from a `universal' IMF. \\citet{Scholz13} and \\citet{Luhman16} find that the cluster NGC 1333 has a greater fraction of sub-stellar objects than IC348, which is interpreted as evidence of enhanced low mass star formation in dense environments. For the most massive and dense local star forming region, the Orion Nebula cluster (ONC), contradictory results have been inferred by different authors. \\citet{DaRio12} find a deficiency of BDs, in direct contrast to a previously inferred enhancement \\citep{Muench02}. Such results might be reconciled by a {bimodal} distribution, as found by \\citet{Drass16}. In this case, the enhancement in occurrence rates at the lowest BD masses may be the result of a distinct formation mechanism, possibly within a protoplanetary disc. \n\nThe distinction between BDs and planets, while motivated by a physical threshold, may not cleanly delineate formation mechanisms \\citep{Chabrier14}. In particular, planets with masses greater than $\\sim 4\\, M_\\mathrm{J}$ may form by gravitational instability rather than core accretion \\citep{Schlaufman18}. Interestingly, the masses of `planets' thought to be responsible for the gaps observed in the dust emission in protoplanetary discs with ALMA \\citep[e.g.][]{HLTau_15, Muller18} can be several Jupiter masses, approaching or exceeding the canonical minimum BD mass \\citep[e.g.][]{Haffert19, Christaens19}. The questions of BD and massive planet occurrence rates may therefore be related. \n\nGlobular clusters are massive populations of stars that remain bound against galactic tides despite their old ages. Having low metallicities and high densities, they represent the present day remnant of formation in a completely different environment to local star forming regions. As such, if the formation of stars or planets is in any way dependent on environment, one should expect to see differences in globular cluster populations with respect to those in the field. \\citet[][hereafter \\citetalias{Gil00}]{Gil00} presented a HST survey of the globular cluster 47 Tuc in search of transiting giant planets. The null result put upper-limits on the frequency of hot Jupiters (with orbits $\\lesssim 10$~days), suggesting a frequency at least an order of magnitude lower than the solar neighbourhood average \\citep[$\\sim 1$~percent --][]{Wright12}. \n\n\nThe potential significance of the dearth of low mass companions of stars in 47 Tuc compounds when one considers the potential for tidal capture in such a dense stellar environment. Tidal capture for close binary formation in globular clusters was suggested by \\citet{Fab75} as a means to explain their high observed X-ray luminosity-to-mass ratios. {This two-body capture mechanism has since been incorporated into Monte Carlo simulations of globular clusters alongside three-body gravitational interactions between `point masses' \\citep{Stodolkiewicz85, Stodolkiewicz86}.} \\citet[][hereafter \\citetalias{Bon03}]{Bon03} pointed out that the same principle also applies to the capture of BDs. This means that if the fraction of BDs that formed in globular clusters is similar to the galactic field, then a significant number of close BD binaries should also exist. Thus, the absence of detected transits in 47 Tuc not only suggests a reduced occurrence rate of hot Jupiters, but also a dearth of BDs. \n\n{In this mini-series of two papers, we explore the dearth of short period sub-stellar companions in 47 Tuc in terms of the expected rates of both BDs (this paper) and hot Jupiters (a second paper). To this end we apply the \\textsc{Mocca} Monte Carlo code \\citep{Hyp13, Giersz13} to accurately compute the theoretical capture and scattering rates over the lifetime of 47 Tuc. In this, the first of the two papers, we focus on the possibility of tidal BD capture in 47 Tuc while a second paper w ill deal with the migration of massive planets to short period orbits. We first discuss the theory of tidal capture in Section~\\ref{sec:tidal_capt_calcs}. We then introduce our Monte Carlo model for 47 Tuc in Section~\\ref{sec:Num_Method}. In Section~\\ref{sec:results} we present the resultant BD capture rates over the lifetime of 47 Tuc, compare to the observational constraints and discuss future observations and generalisation to other globular clusters. Conclusions are summarised in Section~\\ref{sec:conclusions}. }\n\n\n\n\n\n\n\\section{Tidal capture theory}\n\\label{sec:tidal_capt_calcs}\n\\subsection{Tidal capture cross section}\n\\label{sec:tid_cross}\nThe tidal capture condition for a star and BD pair is discussed in detail by \\citetalias{Bon03}; we briefly review the relevant equations here. During the close passage between stars, the orbital energy can be dissipated by non-radial oscillations within the stellar interiors \\citep{Rob68}. It follows that if the passage is sufficiently close, the tidal dissipation can result in capture and the formation of a tight binary. \\citet{Fab75} applied this mechanism to explain low mass X-ray binaries found in globular clusters. We consider a primary star of radius $R_*$, mass $m_*$. Then for a periastron distance $a_\\mathrm{p}$ during an encounter with a much smaller (point) mass $m$, the condition for capture can be approximated \\citep{Fab75}:\n\\begin{equation}\n\\label{eq:capt_rad}\n \\frac{a_\\mathrm{p}}{R_*} < \\frac{a_\\mathrm{capt}}{R_*} = \\left[ \\frac{Gm_*}{R_* v_\\infty^2} q (1+q) \\right]^{1\/6}\n\\end{equation} where $q=m\/m_*$, for secondary mass $m$, and $v_\\infty$ is the relative velocities of the two stars at infinite separation, $a_\\mathrm{capt}$ is the capture radius. The point mass approximation is justified since tides can only be excited in a much smaller secondary if the impact parameter is within the collisional cross section. Equation~\\ref{eq:capt_rad} remains valid for low mass main-sequence primaries \\citep[$n=3\/2$ polytropes,][]{Lee86}. \n\nThe capture radius must also exceed the sum of the radii of the two interacting bodies, otherwise the objects would collide (i.e. periastron distance $a_\\mathrm{p}> R_*+R_\\mathrm{bd}$, where $R_\\mathrm{bd}$ is the BD radius). We will hereafter assume that all BDs have $R_\\mathrm{bd}=0.1 \\, R_\\odot$. For a star and BD pair the capture cross section is:\n\\begin{equation}\n\\label{eq:sigma_capt}\n \\sigma_\\mathrm{capt}=\\begin{cases}\n \\sigma_\\mathrm{capt}' - \\sigma_\\mathrm{coll}\\, & \\sigma_\\mathrm{capt}' >\\sigma_\\mathrm{coll}\\\\\n 0 \\, &\\rm{otherwise}\n \\end{cases},\n\\end{equation} where $\\sigma_\\mathrm{coll}$ is the collisional cross section (including gravitational focusing) and \n\\begin{equation}\n\\label{eq:sigmacapt_dash}\n\\sigma_\\mathrm{capt}' = \\pi a_\\mathrm{capt}^2 \\left[ 1 + \\frac{2Gm_*(1+q)}{v_\\infty^2 a_\\mathrm{capt}}\\right]\n\\end{equation} is the capture cross section if collisions are ignored. \n\nWithin the \\textsc{Mocca} framework, we implement the tidal capture scenario in the same way as stellar collisions \\citep{Freitag02}. In brief, this involves looping over all stars within a local subset \\citep[`zone' -- see][]{Gie98}, and finding a corresponding BD pair at random. {We compute the local number density $n_\\mathrm{bd}$ of BDs in the same way as stars in the \\textsc{Mocca} framework. In brief, this involves finding a number of the closest objects in cluster radius ($r$) space and normalising by the minimum spherical shell volume that encloses them.} The probability of capture between the pair is:\n\\begin{equation}\n P_{\\mathrm{capt}} = n_\\mathrm{bd} \\sigma_\\mathrm{capt} v_\\infty \\Delta t, \n\\end{equation} for time-step $\\Delta t$. In this way, due to the normalisation by the BD density, it is only necessary to loop over all stars and not both stars and BDs. In the case of capture, a star-BD binary is produced with a circular orbit (assuming a short circularisation time-scale) and semi-major axis $a_\\mathrm{bd} = 2 a_\\mathrm{p}$ \\citep{Mardling96}. {To determine $a_\\mathrm{p}$ for a given encounter, we first draw the encounter cross section uniformly between $\\sigma_\\mathrm{coll}$ and $\\sigma'_\\mathrm{capt}$, then assign the corresponding $a_\\mathrm{p}$. Apart from for tidal capture scenarios, BDs are treated as main sequence stars in \\textsc{Mocca}, with associated collision probabilities and dynamical interactions.}\n\n\n\\subsection{Capture rates}\n\\label{sec:theory_rates}\n\n{As discussed in Section~\\ref{sec:post-process}, it is useful to not only compute the cross section and capture probability for BD-star pairs in the Monte Carlo simulation but also post-process the encounter probability for individual stars. This allows us to compute the scaling of the capture probability over the physical parameter space, where the Monte Carlo may only yield a small number of encounters that result in large uncertainties. Therefore we compute the encounter rates as a function of stellar and environmental parameters in this section. We make an analytic estimate assuming no collisions to give an intuition as to how the capture rate scales (Section~\\ref{sec:analytic_approx}). We then compute the capture rates with star-BD collisions included, demonstrating how these collisions reduce the capture rates (Section~\\ref{sec:full_calc}). } \n\n\\begin{figure}\n \\centering\n \\subfloat[\\label{subfig:sigvar}Encounter cross sections with relative speed]{\\includegraphics[width= 0.8\\columnwidth]{sigcolcapt.pdf}}\\\\\\vspace{-10pt}\n \\subfloat[\\label{subfig:dgcdv}Differential capture rate]{\\includegraphics[width= 0.8\\columnwidth]{dgcaptdv.pdf}}\\\\ \\vspace{-10pt}\n \\subfloat[\\label{subfig:Gamma_capt}Overall per star capture rates ]{\\includegraphics[width= 0.8\\columnwidth]{gamma_capt.pdf}}\\\\\\vspace{-3pt}\n \\caption{Computations of the key quantities in determining the tidal capture rates of BDs in a given stellar environment. Figure~\\ref{subfig:sigvar} shows the effective cross sections of collision (dotted lines) and capture (solid lines) for varying relative speeds at infinite separation. Figure~\\ref{subfig:dgcdv} shows the corresponding differential capture rates (equation~\\ref{eq:Gamma_capt}) for different velocity dispersions. Figure~\\ref{subfig:Gamma_capt} shows the integrals of this differential across all velocities (solid lines) compared to the approximation in equation~\\ref{eq:approx_captrate} (dotted lines). In all cases, the lines are coloured by the assumed stellar properties. Results in Figures~\\ref{subfig:dgcdv} and~\\ref{subfig:Gamma_capt} are shown for BD number density $n_\\mathrm{bd}\/10^6$~pc$^{-3} = n_{\\mathrm{bd},6} = 1$ and scale linearly with this value. }\n \\label{fig:Gammacapt_sigv}\n\\end{figure}\n\n\\subsubsection{Estimated encounter rate by \\citetalias{Bon03}}\n\n {Given a local velocity dispersion and BD density, the instantaneous tidal capture rate for an individual star can be estimated. This rate $\\Gamma_\\mathrm{capt}$ is the rate at which neighbours pass within the effective cross section for capture, equation~\\ref{eq:sigma_capt}. In general, if all objects have mass $m$, velocity dispersion $\\sigma_v$ and density $n$ then the encounter rate within a given radius $a_\\mathrm{enc}$ can be written \\citep{Binney08}:}\n \\begin{equation}\n \\Gamma_\\mathrm{enc} = 16\\sqrt{\\pi} \\cdot n \\sigma_v a_\\mathrm{enc}^2 \\cdot \\left( 1+\\frac{G m}{2\\sigma_v^2 a_\\mathrm{enc}}\\right).\n \\end{equation}{In the gravitationally focused regime, the second term in brackets dominates and $\\Gamma_\\mathrm{enc} \\propto a_\\mathrm{enc}\/\\sigma_v$. If $a_\\mathrm{enc} = a_\\mathrm{capt}- (R_*+R_{\\mathrm{bd}})$ is not strongly dependent on $v_\\infty$, then for BD density $n_\\mathrm{bd}$ the BD capture rate for a star of mass $m_*$ is:\n\\begin{multline}\n\\label{eq:Gammacapt_Bonnell}\n \\Gamma_\\mathrm{capt}^\\mathrm{BCB+03} \\approx 1.4 \\times 10^{-4} \\, \\left( \\frac{n_\\mathrm{bd}}{10^6\\, \\rm{pc}^{-3}}\\right) \\times \\\\ \n \\times \\left( \\frac{\\sigma_v}{10\\, \\rm{km\\, s}^{-1}}\\right)^{-1} \\frac{a_\\mathrm{enc}}{1\\, R_\\odot} \\frac{m_*}{1\\, M_\\odot} \\,\\rm{Myr},\n\\end{multline}where \\citetalias[][]{Bon03} apply this expression with fixed $v_\\infty = \\sigma_v = 10$~km\/s. Adopting equation~\\ref{eq:Gammacapt_Bonnell} and assuming $v_\\infty = \\sigma_v$ to obtain $a_\\mathrm{enc}$ is accurate when the $\\sigma_v$ is sufficiently small -- i.e. when $\\sigma_\\mathrm{capt}' \\gg \\sigma_\\mathrm{coll}$ for $v_\\infty \\approx \\sigma_v$. However, both $\\sigma'_\\mathrm{capt}$ and $\\sigma_\\mathrm{coll}$ are dependent on the relative velocities of the pairs, stellar mass and radius in different ways, such that understanding the scaling of $\\sigma_\\mathrm{capt}$ on these properties is non-trivial when $\\sigma_\\mathrm{capt}' \\approx \\sigma_\\mathrm{coll}$ for $v_\\infty \\approx \\sigma_v$. It is therefore unclear what kind of encounters dominate the overall capture rate in general and we will find that $\\sigma_v$ significantly exceeds $10$~km\/s in our dynamical model for 47 Tuc. For this reason, we must generally integrate over the full differential capture rate, including the capture cross section. }\n\n\n\\subsubsection{Analytic approximation without collisions}\n\\label{sec:analytic_approx}\nThe differential capture rate as a function of the velocity at infinity $v_\\infty$ is:\n\\begin{equation}\n\\label{eq:Gamma_capt}\n \\mathrm{d} \\Gamma_\\mathrm{capt} = \\sigma_\\mathrm{capt}(v_\\infty; q, R_*) \\,n_{\\rm{bd}} \\, v_\\infty g(v_\\infty; \\sigma_v) \\,\\rm{d}v_\\infty \n\\end{equation}where\n\\begin{equation}\n g(v_\\infty; \\sigma_v) = \\frac{v_\\infty^2}{2\\sqrt{\\pi} \\sigma_v^3} \\exp \\left(\\frac{-v_\\infty^2}{4\\sigma_v^2} \\right)\n\\end{equation} is the Maxwell-Boltzmann distribution, or the relative asymptotic speed $v_\\infty$ distribution for dispersion $\\sigma_v$. In the limit $\\sigma_\\mathrm{capt}\\gg \\sigma_\\mathrm{coll}$ (small $v_\\infty$, large $m_*$) for the dominant capture scenarios, equation~\\ref{eq:Gamma_capt} can be integrated over all velocities to yield the analytic upper-limit to the capture rate:\n\\begin{multline}\n\\label{eq:approx_captrate}\n \\Gamma_\\mathrm{capt} < 3.7 \\times 10^{-7} \\left[\\left(\\frac{R_*}{1\\,R_\\odot}\\right)^5 \\frac{m_*}{1\\,M_\\odot} \\frac{\\sigma_v}{10\\,\\rm{km \\, s}^{-1} }q (1+q) \\right]^{1\/3}\\times \\\\\n \\times \\left(1+ \\phi_\\mathrm{grav} \\right) \\cdot \\left( \\frac{n_{\\rm{bd}}}{10^6\\, \\rm{pc}^{-3}}\\right) \\, \\rm{Myr}^{-1},\n\\end{multline} where \n\\begin{align}\n\\nonumber\n \\phi_{\\rm{grav}} &\\approx 423\\, q^{-1\/6}(1+q)^{5\/6} \\quad\\times \\\\ &\\qquad\\times \\left(\\frac{m_*}{1\\,M_\\odot}\\right)^{5\/6} \\left(\\frac{R_*}{1\\,R_\\odot}\\right)^{-5\/6} \\left(\\frac{\\sigma_v}{10\\, \\rm{km\\,s}^{-1}} \\right)^{-5\/3}\n\\end{align} is the gravitational focusing factor. While these expressions are approximate, they highlight three things:\n\\begin{enumerate}\n \\item The relevant encounters are in practice always gravitationally focused ($\\phi_\\mathrm{grav}\\gg1$), independently of the type of star under consideration, where we assume $R_*\\propto m_*^\\beta$ with $\\beta\\sim 1$. The velocity dispersion required for $\\phi_\\mathrm{grav}\\sim 1$ is $\\sigma_v \\sim 100$~km~s$^{-1}$. \n \\item The capture rate scales steeply with the mass (radius) of the star: $\\Gamma_\\mathrm{capt} \\propto q^{1\/6} R_*^{5\/6} m_*^{7\/6} \\propto m_*^{1 +5\\beta\/6}$. This steep scaling highlights the importance of the mass function in assessing the total number of encounters. In particular, for mass function $\\xi$ we have $\\Gamma_\\mathrm{capt}\\cdot \\xi \\,\\mathrm{d}m_* \\propto m_*^{-\\gamma}\\,\\mathrm{d}m_*$ for $\\gamma \\lesssim 1$, such that the integral diverges for large $m_*$. The overall capture rate is therefore initially dependent on the choice of maximum stellar mass $m_\\mathrm{max}$.\n \\item The capture time-scale ($\\propto 1\/\\Gamma_\\mathrm{capt}$) scales super-linearly with the velocity dispersion ($\\Gamma_\\mathrm{capt} \\propto \\sigma_v^{-4\/3}$). The temporal and spatial evolution of the local velocity dispersion within a globular cluster is therefore an important factor in determining capture frequency.\n\\end{enumerate}\n\n\n\\subsubsection{Full calculation with collisions}\n\\label{sec:full_calc}\n\nThe stellar mass and velocity dispersion become even more important when one numerically integrates the full expression for equation~\\ref{eq:Gamma_capt}. {We show the relevant quantities in computing the capture rate in Figure~\\ref{fig:Gammacapt_sigv}. The comparison of the cross sections for capture and collisions are in Figure~\\ref{subfig:sigvar}. The collision cross section exceeds the cross section for capture above some relative speed $v_\\infty$, which increases with stellar mass. The effect of this on the differential encounter rate for given velocity dispersion $\\sigma_v$ is shown in Figure~\\ref{subfig:dgcdv}, the form of which is non-trivially dependent on the stellar properties. }\n\nThe analytic approximation in the limit $\\sigma_\\mathrm{coll} \\ll \\sigma_\\mathrm{capt}$ (equation~\\ref{eq:approx_captrate}) is compared with this numerical integration in Figure~\\ref{subfig:Gamma_capt} as a function of the velocity dispersion. As $\\sigma_v$ becomes large, from equation~\\ref{eq:capt_rad} we have that $a_\\mathrm{capt}\/R_*$ becomes small for all possible encounters, such that neglecting collisions is not possible. The result is a (stellar mass dependent) steep decline in the capture rate with $\\sigma_v$, much steeper than the analytic approximation of $\\Gamma_\\mathrm{capt}\\propto \\sigma_v^{-4\/3}$. Since the approximation is only valid for smaller $\\sigma_v$ than is typical for globular clusters, we will hereafter always adopt the full numerical integration when estimating capture rates. \n\nThe deviation from our approximation particularly affects the lower mass stars, for which gravitational focusing becomes weaker at a lower velocity dispersion relative to higher mass stars. The decrease in capture efficiency for low mass stars is precipitous for $m_*\\lesssim 0.5\\,M_\\odot$, such that the stellar mass function must be considered in order to compute global capture rates. We further consider how the evolving mass function alters the overall capture rates in Appendix~\\ref{app:massfunc}, where we justify adopting $m_* = 0.7\\,M_\\odot$ to approximate the global capture rate over the lifetime of 47 Tuc. \n\n\n\\section{Dynamical modelling}\n\n\\label{sec:Num_Method}\n\n\n\\subsection{Monte Carlo simulations}\n\nSimulating the dynamical evolution of globular clusters directly using $N$-body calculations over their $\\sim 10$~Gyr evolution is not computationally practicable. For this reason, the \\textsc{Mocca} code \\citep{Gie98, Gie01} has been developed as a {Monte Carlo} approach to statistically computing the evolution of massive, dense stellar clusters by solving the Fokker--Planck equation \\citep[see also][]{Sto82}. This approach allows a fast and accurate calculation of the dynamical evolution of the stellar population in a globular cluster over its lifetime. The added bonus of using this code is that it has already been applied to model the evolution of 47 Tuc \\citep[][hereafter \\citetalias{Gie11}]{Gie11}. We are therefore able to adopt the parameters obtained in this previous modelling effort. Where appropriate, we make similar comparisons to observational constraints as \\citetalias{Gie11}. \\textsc{Mocca} has since been updated to incorporate the \\textsc{Fewbody} code \\citep{Fregeau04, Fregeau12} into an improved prescription for interactions between multiple systems, as described by \\citet{Hyp13}. Stellar evolution modules by \\citet{Hurley00, Hurley02} are used to compute single star and binary evolution. We do not include tidal forces between a star and a companion in this evolution. \n\n\\subsection{Initial conditions for 47 Tuc}\n\n\\label{sec:ICs}\n\\begin{figure*}\n \\subfloat[\\label{subfig:surfmag}Surface brightness]{\\includegraphics[width= 0.47\\textwidth]{surfbright.pdf}}\n \\subfloat[\\label{subfig:los_vdisp}Line of sight velocity dispersion]{\\includegraphics[width= 0.47\\textwidth]{los_vdisp.pdf}}\n \n \\caption{Observational constraints on the dynamical properties of 47 Tuc compared to the results of our \\textsc{Mocca} (Monte Carlo) model with the parameters listed in Table~\\ref{table:dyn_mods}. Figure~\\ref{subfig:surfmag} shows the visible surface brightness profile of the Monte Carlo model (red circles) compared to the observed profile found by \\citet[][black crosses]{Tra95}. The line-of-sight velocity dispersion is compared in Figure~\\ref{subfig:los_vdisp}, where the Monte Carlo results are again red circles. The relative line of sight velocities measured by \\citet{Gebhardt95} are shown by faint points and the inferred dispersion shown as a black line. The sampling uncertainties are indicated by dashed lines. }\n \\label{fig:47Tuc_obs}\n\\end{figure*}\n\n\\begin{table}\n\\centering \n \\begin{tabular}{c c c } \n \\hline\n Parameter & Property & Value \\\\ [0.5ex] \n \\hline\n$N_*$ & Number of stars & $2 \\times 10^6$ \\\\\n$N_\\mathrm{bd}$& Number of BDs & $2\\times 10^6$ \\\\\n$N_\\mathrm{bin}$ & Number of binaries & $4.4\\times10^4$ \\\\ \n$N_\\mathrm{pl}$ & Number of planets & $2 \\times10^4$ \\\\ \n$W_0$ & Central concentration & $7.5$ \\\\\n$m_\\mathrm{br}\/M_\\odot$ & Break mass & 0.8 \\\\\n$m_\\mathrm{max}\/M_\\odot$ & Max. mass & 50\\\\\n$\\alpha_1$ & IMF slope $mm_\\mathrm{br}$ & 2.8\\\\\n$T_\\mathrm{age}\/$Gyr & Age & $12$ \\\\\n$Z\/Z_\\odot$ & Metallicity (dex) & -0.6 \\\\\n$e_0$ & Planet eccentricity & 0.9 \\\\ \n$a_0\/$au & Planet semi-major axis & $5$\\\\ \n \\hline\n\\end{tabular}\n\\caption{Initial condition parameters used for the Monte Carlo globular cluster model discussed in the text. Where appropriate, choices are made to match the model of \\citetalias{Gie11} for 47 Tuc. BDs and planets are the same except that planets are initially companions to stars.}\n\\label{table:dyn_mods}\n\\end{table}\n\nThe initial conditions that we adopt are motivated by the findings of \\citetalias{Gie11}, who reproduced the key observable properties of 47 Tuc. The main parameters are summarised in Table~\\ref{table:dyn_mods}. The binary population are drawn from a log-uniform distribution between $1{-}100$~au. {The initial conditions include a number of choices that are somewhat artificial (such as the low maximum stellar mass, small binary fraction and no mass fallback for black hole formation). These choices were invoked to reproduce the unusual surface brightness and velocity dispersion profiles. A low binary fraction is also convenient in our context, because we compute tidal capture rates only between BDs and single stars. More recent models incorporating a higher binary fraction and maximum stellar mass can reproduce the frequency of special objects (e.g. black holes binaries and pulsars) and central surface brightness, but do not presently reproduce the observed surface brightness profile (A. Askar -- private communication). The most important property for computing the BD capture rate is the stellar density profile. We therefore retain the parameters of \\citetalias{Gie11} with which we find good agreement with the observed density and velocity dispersion profiles (see Section~\\ref{sec:47Tuc_obs_comp}). Provisional checks using the unpublished alternative models yield BD capture rates similar to those we obtain with our fiducial model. However, we emphasise initial conditions that can reproduce the properties of globular clusters are degenerate. Although models that yield a similar density profile probably yield similar capture rates (see discussion of caveats in Section~\\ref{sec:caveats}), an extensive parameter study investigating these choices is outside of the scope of this work.} \n\n\n\nWe additionally include a BD population, with equal numbers as the stars. The masses of the BDs have initial masses $m_\\mathrm{bd} = 0.079\\, M_\\odot$, to ensure that their masses are lower than that of the least massive stars. In the case that an object has a mass that exceeds $0.08 \\, M_\\odot$ (for example, via a collision\/merger), then the object is no longer defined as a BD. The initial spatial distribution of BDs is assumed to be the same as the stellar population (i.e. no primordial mass segregation).\n\nFinally, we add a population of `migrating planets' around 1 percent of the initial stellar population. {The orbital evolution of this population will be considered in Paper II, but are not relevant in this work.} We adopt the same mass as the BDs for simplicity. While this results in a greater mass ratio $q$ than for planets, the scattering cross section for binaries is only weakly dependent on the mass ratio, particularly for $q\\lesssim 0.1$ \\citep{Fregeau04}. We initialise all of the planet orbits to have eccentricity $e=0.9$ and semi-major axes $a = 5$~au, reflecting a Jupiter analogue with high eccentricity. Planets are paired with stars drawn from the same IMF as single stars. {Because the companion (planet) population is a small fraction of the stellar population, as well as low mass and with small semi-major axis compared to the binaries, this population has a negligible influence on the overall evolution of the \\textsc{Mocca} simulation (confirmed by performing runs without them). We do not include these systems in the tidal capture Monte Carlo routine, although this too has a negligible effect on the overall capture rate due to the low companion fraction. They are treated as binaries within the \\textsc{Mocca} framework, undergoing single-binary and binary-binary interactions integrated with \\textsc{Fewbody}. We will only refer to this population again in Paper II. }\n\n\\subsection{Comparison with observed properties of 47 Tuc}\n\\label{sec:47Tuc_obs_comp}\nWe wish to ensure that the model approximately reproduces the key physical properties of 47 Tuc at its present age. These are the stellar density and velocity dispersion profile. \\citetalias{Gie11} fitted their models to the surface brightness profile as measured by \\citet{Tra95} and the line-of-sight velocity dispersion inferred from the measurements of \\citet{Gebhardt95}. For direct comparison, we perform the same comparisons for our model. \n\nThe $V$-band surface magnitude profile after integrating the model for 12 Gyr is shown in Figure~\\ref{subfig:surfmag}, adopting a distance of $4$~kpc. The profile is calculated by averaging over the $V$-band luminosity contribution of concentric shells of stars for distance $d$ from the cluster centre:\n\\begin{equation}\n\\label{eq:Vband_surf}\n \\Sigma_V(d) =\\sum_{r_i>d}\\frac{L_V}{2\\pi r_i^2} \\frac{r_i}{\\sqrt{r_i^2-d^2}}.\n \\end{equation}\n We then convert this to a surface magnitude by the expression:\n\\begin{equation}\n \\mu_V = V_\\odot -2.5\\log \\Sigma_{V}' + A_V,\n\\end{equation}where $V_\\odot = 4.80$ is the solar $V$-band magnitude and $\\Sigma_{V}' $ is $\\Sigma_{V} $ in units of solar $V$-band luminosity per square arcsecond. {We assume small visual extinction $A_V$. Commonly, the \\citet{Harris96} value of $E(B-V) = 0.04$ is used \\citep[although smaller value $E(B-V)=0.024$ may also be adopted --][]{Crawford75}, to give $A_V \\approx 0.12$ for relative visibility $R_V = 3.1$.} Despite the updated version of \\textsc{Mocca} and the inclusion of BDs compared to \\citetalias{Gie11}, we find reasonable agreement between the model and the observed profile obtained by \\citet{Tra95} similarly to \\citetalias{Gie11}. \n\nThe line-of-sight velocity dispersion profile can be computed in an analogous fashion to the surface brightness and is shown in Figure~\\ref{subfig:los_vdisp}. The dispersion at projected separation $d$ from the centre is the contribution of the projected contributions of the radial and tangential velocities ($v_r$ and $v_{\\rm{t}}$ respectively):\n\\begin{equation}\n \\sigma_{v,\\rm{los}}^2(d)= \\frac{1}{n_{d}}\\sum_{r_i>d} \\frac{d}{r_i\\sqrt{r_i^2-d^2}}\\left[v_r^2\\frac{r_i^2-d^2}{r_i^2} + \\frac {v_{\\rm{t}}^2} 2 \\frac{d^2}{r_i^2} \\right] , \n\\end{equation} where \n\\begin{equation}\n n_d= \\sum_{r_i>d} \\frac{d}{r_i\\sqrt{r_i^2-d^2}}.\n\\end{equation}The corresponding observed dispersion can be extracted from the measured line-of-sight velocities by computing the dispersion relative to the mean velocity, binned by separation from the cluster centre. The comparison in Figure~\\ref{subfig:los_vdisp} shows that the two dispersions are similar across the separations with observational constraints. Given that both the surface density and velocity dispersion profiles are comparable to the observational constraints, we adopt this model without further (computationally expensive) parameter study. \n\n\n\\subsection{Stellar density and velocity evolution}\n\nThe main quantities of interest for computing the rate of close encounters in a stellar cluster are the local number density and velocity dispersion. We post-process the output of our Monte Carlo integration to track the density and velocity dispersion evolution over the 12 Gyr lifetime of 47 Tuc. We divide the stars by cluster radii into $30$ log-uniformly spaced bins between $10^{-1.5}$ and $10^1$~pc, then normalise the number in each shell by the volume to yield the density. For velocity, we perform a similar binning but then compute the dispersion $\\sigma_v$ in the one dimensional velocities: $\\sqrt{ v_r^2+v_\\mathrm{t}^2}$, where $v_r$ is the velocity in the radial direction, and $v_\\mathrm{t}$ is the tangential component (combined azimuthal and polar). \n\n\\subsubsection{Density evolution}\n\nWe show the density profile evolution in Figure~\\ref{fig:n_evol}, separated into stars (dashed) and BDs (dotted). The initial core stellar density distribution is $n_*\\sim 10^6$~pc$^{-3}$ within $\\sim 0.5$~pc, and stellar densities $\\gtrsim 10^5$~pc$^{-3}$ are retained for several Gyr. However, this is not true for the BD population. During relaxation, dynamical interaction leads to energy equipartition, resulting in lower masses being pushed to regions of a shallower gravitational potential; this is known as mass segregation \\citep{Binney08}. By this process highest density regions at the cluster centre are quickly vacated of BDs to yield densities $n_\\mathrm{bd}\\lesssim 10^5$~pc$^{-3}$ within $\\sim 1$~Gyr. This will have significant consequences on the efficiency of tidal capture over the lifetime of 47 Tuc (see Section~\\ref{sec:bd_capt}). \n\n\n\\begin{figure} \n \\centering\n \n \\includegraphics[width=\\columnwidth]{density_evol.pdf}\n \\caption{Density profile evolution for stars (dashed) and BDs (dotted) in our \\textsc{Mocca} model for the dynamical evolution of 47 Tuc. The lines are shown every $500$~Myr, coloured by the time in the simulation. Both stars and BDs initially have the same density profile.}\n \\label{fig:n_evol}\n\\end{figure}\n\n\\subsubsection{Velocity dispersion evolution}\n\\label{sec:sigmav_evol}\n\n{The evolution of the velocity dispersion profile is shown in Figure~\\ref{fig:sigmav_evol}. In this case the velocity dispersion across the majority of the cluster decreases as it relaxes. The very inner region retains a high velocity dispersion, which is due to the fact that the majority of trajectories passing through this region are at the pericentre of an eccentric orbit. The physical (three dimensional) dispersion is a factor $\\sqrt{3}$ greater than the one dimensional dispersion. However, the line-of-sight dispersions inferred from radial velocity measurements and our reconstructed `observation' of the model ($\\sim 10{-}12$~km~s$^{-1}$ in the central regions; Section~\\ref{sec:ICs}) are lower again by a factor of order unity. This is due to projection effects. The stellar density profile means that the majority of stars are a few parsec from the centre in three dimensions. When one infers the dispersion of radial velocities at projected (two dimensional) separations smaller than this, the measurements are mostly made for outer, lower velocity stars that fall along the line-of-sight. Thus the apparent dispersion is an underestimate of the physical one dimensional dispersion in the central regions. }\n\n\nAs a result of these considerations, the inner physical three dimensional dispersion is much larger in the highest density regions than the fiducial $10$~km~s$^{-1}$ assumed by \\citetalias{Bon03}. In Section~\\ref{sec:theory_rates} we show that the capture rate decreases much steeper than linearly with $\\sigma_v$. These two findings demonstrate why the approach we take is necessary in computing accurate capture rates over the lifetime of a globular cluster. A velocity dispersion of $\\sigma_v= 30$~km~s$^{-1}$ can result in order of magnitude decreases in the capture rate compared to $\\sigma_v=10$~km~s$^{-1}$, depending on the stellar mass. The full, local velocity dispersion evolution is therefore a necessary ingredient in the calculations we perform. \n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{sigmav_evol.pdf}\n \\caption{Velocity dispersion profile evolution in the \\textsc{Mocca} model for the dynamical evolution of 47 Tuc. Lines are coloured by time in the simulation, each separated by $500$~Myr.}\n \\label{fig:sigmav_evol}\n\\end{figure}\n\n\n\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\subsection{Overall capture rate}\n\\label{sec:bd_capt}\n\\label{sec:overall_capt}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{captrate_evol.pdf}\n \\caption{Top: The instantaneous rate $\\Gamma_\\mathrm{capt}$ of tidal BD capture for a star of mass $m_*=0.7\\, M_\\odot$ with the mass-radius relation given by equation~\\ref{eq:approx_mr} at a given cluster radius within 47 Tuc. Contours are computed by integrating over equation~\\ref{eq:Gamma_capt} given the BD number density and velocity dispersion shown in Figures~\\ref{fig:n_evol} and~\\ref{fig:sigmav_evol} respectively. {Bottom: The same countours but now weighted by $4\\pi r^2 n_*$, giving the estimated rate of overall tidal capture per unit radius. This is an approximation for the integrand in equation~\\ref{eq:totcaptrate}. }}\n \\label{fig:Gammacapt_evol}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{tidal_capts.pdf}\n \\caption{Cumulative number of tidal BD captures through the lifetime of 47 Tuc. The black line shows the result obtained directly from the Monte Carlo calculation. The red line is an estimate using the local stellar\/BD density and velocity dispersion with equation~\\ref{eq:Gamma_capt}, assuming $m_*=0.7\\,M_\\odot$ for all stars (see discussion in Appendix~\\ref{app:massfunc}). The poor agreement after $\\sim 1$~Gyr is due to {stellar} mass segregation (see text for details).}\n \\label{fig:MCvest_capts}\n\\end{figure}\n\nThe total number of BD captures in the Monte Carlo simulation of 47 Tuc is 377. In order to interpret this directly computed capture rate, we take the density and velocity dispersion profiles from our models to approximate the global capture rates by post-processing {the density and velocity dispersion profiles to obtain the encounter rates according to equation~\\ref{eq:Gamma_capt}}. {To perform this calculation, we need to adopt an expression that encapsulates the stellar mass averaged encounter rate:\n\\begin{equation}\n \\hat{\\Gamma}_\\mathrm{capt} = \\int_{m_{\\mathrm{min}}}^{m_\\mathrm{max}} \\, \\mathrm{d} m_* \\, \\xi(m_*) \\Gamma_\\mathrm{capt} (m_*),\n\\end{equation}which strictly requires calculating the temporal and spatial evolution of the mass function $\\xi$. However, we show in Appendix~\\ref{app:massfunc} that if the mass function remains constant then after a short time-scale ($\\sim 100$~Myr) the removal of the most massive stars results in $ \\hat{\\Gamma}_\\mathrm{capt} \\approx \\Gamma_\\mathrm{capt} (0.7\\,M_\\odot)$, which remains true over the majority of the lifetime of 47 Tuc. This approximates the global temporal evolution of the mass function, but not the spatial variation. We will see that the assumption of a spatially homogeneous mass function is an important omission.}\n\nIntegrating over equation~\\ref{eq:Gamma_capt} yields a local capture rate over the dynamical history of 47 Tuc. The results of this calculation are shown in the top panel of Figure~\\ref{fig:Gammacapt_evol}. This in turn can be integrated to give a global capture rate:\n\\begin{equation}\n\\label{eq:totcaptrate}\n\\dot{N}_\\mathrm{capt}= \\int_0^\\infty\\,\\mathrm{d}r \\, 4\\pi r^2 \\, n_*(r) \\hat{\\Gamma}_\\mathrm{capt} (r),\n\\end{equation}where we adopt the stellar mass averaged capture rate $ \\hat{\\Gamma}_\\mathrm{capt} \\approx \\Gamma_\\mathrm{capt} (0.7\\, M_\\odot)$ (see Appendix~\\ref{app:massfunc}). \n\nThe integrand of equation~\\ref{eq:totcaptrate} is shown in the bottom panel of Figure~\\ref{fig:Gammacapt_evol}. The evolution of the capture rate reflects the fact that in the early stages BDs occupy the central, high density regions, but are quickly evacuated to the outer regions as the cluster becomes mass segregated. During the segregation, the capture rate profile quickly transitions from being centrally concentrated to relatively flat with radius. In general, the capture rates are slightly lower than the estimates by \\citetalias{Bon03} due to the considerations discussed in Section~\\ref{sec:theory_rates}; principally the strong dependence of capture efficiency on the stellar mass and local velocity dispersion. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{mmed_evol.pdf}\n \\caption{Median stellar mass binned by radius from the cluster centre over the lifetime of 47 Tuc in our Monte Carlo model. Lines are coloured by time in the simulation, each separated by 500~Myr.}\n \\label{fig:mmed_profile}\n\\end{figure}\n\nIn Figure~\\ref{fig:MCvest_capts} we compare the result of the estimated encounter rate computed using equation~\\ref{eq:totcaptrate} {(assuming a constant stellar mass function over space and time)} with the number of captures obtained directly from the Monte Carlo simulation. The rate of capture is initially well reproduced by our estimate, but after approximately $1$~Gyr we overestimate the global rate of captures. This highlights a problem with approximating the encounter rates: we have not accounted for the segregation of \\textit{stellar} mass throughout the cluster (only the BDs). Figure~\\ref{fig:mmed_profile} shows the median stellar mass in radial bins over the dynamical evolution of our Monte Carlo model. The model is not initially segregated, which is why we have good agreement between the Monte Carlo and the post-processed approximation. However, a gradient in stellar masses quickly emerges as mass segregation operates. The result is that lower mass stars preferentially occupy the same regions as the BDs. Capture for low mass stars is inefficient due to the steep decline in the cross section with decreasing stellar mass when the local velocity dispersion is large (Section~\\ref{sec:theory_rates}). Hence the capture rates are further suppressed by the separation of the BDs and high mass stars by which they can be efficiently captured. \n\n\\subsection{Semi-analytic encounter rate calculations}\n\\label{sec:post-process}\n\n{The Monte Carlo approach we describe in Section~\\ref{sec:tid_cross} has the benefit that we can generate a realistic number of capture or scattering events across a complex parameter space. However, there are benefits to complementing the Monte Carlo with semi-analytical estimates. In particular, this allows us to more easily understand the scaling of the results with the stellar parameters and cluster properties. In the Monte Carlo calculation, the number of events may be low in some regions of parameter space (e.g. stellar mass and position), which results in large uncertainties in recovering the probability from sampling. To construct probability functions for encounter rates, a better approach is to take a subset of stars at the end of the simulation and track their encounter rate throughout their lifetime. This has the added benefit that we can use the analytic expressions to scale our results based on assumed physical properties.}\n\nTo recover the encounter rate evolution we must integrate over orbits which are much shorter than is possible to temporally resolve with the output time-step. At each snapshot we therefore recover the orbital solution by first fitting an approximate analytic double power-law density profile:\n\\begin{equation}\n\\label{eq:rho_pot}\n \\rho_* = \\frac{M_\\mathrm{s}}{4\\pi a_\\mathrm{s}^3} (r\/a_\\mathrm{s})^{-\\alpha}(1-r\/a_\\mathrm{s})^{\\alpha-\\beta}\n\\end{equation} to the stellar mass density, where $M_\\mathrm{s}$, $a_\\mathrm{s}$, $\\alpha$ and $\\beta$ are fitting constants. We then construct the corresponding spherically symmetric potential using the \\texttt{TwoPowerSphericalPotential} class of \\textsc{Galpy}\\footnote{\\url{http:\/\/github.com\/jobovy\/galpy}} \\citep{Bovy15}. {Although orbits in the potential described by equation~\\ref{eq:rho_pot} are not closed, since the density and velocity dispersion profiles are spherically symmetric we are only interested in the radial oscillations of the star with respect to the respect the centre of mass of the cluster. We therefore define the period $P_\\mathrm{orb}$ for the star to make a single epicycle. Then the} orbitally averaged {capture} rate at the specified time-step is:\n\\begin{equation}\n\\label{eq:orbavg}\n \\langle \\Gamma_\\mathrm{capt} \\rangle (\\Theta(t_\\mathrm{step}))= \\frac{1}{P_\\mathrm{orb}} \\int_{t_\\mathrm{step}}^{t_\\mathrm{step}+P_\\mathrm{orb}} \\,\\mathrm{d}t \\, \\Gamma_\\mathrm{capt}(\\Theta(t)),\n\\end{equation}where $t$ is the time coordinate, $t_\\mathrm{step}$ is the time of the snapshot and all other pertinent parameters are enclosed in $\\Theta$. In practice, if the time-step between updating orbital solution $\\Delta t10^5\\,M_\\odot$ are summarised in Table~\\ref{table:GC_scaled} and represented in Figure~\\ref{fig:GC_plots}. {We have listed the approximate number of stars $N_{*, 0.52-0.88}$ in the mass range surveyed by \\citetalias{Gil00} by integrating the mass function we assume for the 47 Tuc model, truncated above $0.88 \\, M_\\odot$ in the relevant range.} A number of clusters have significantly higher densities than 47 Tuc, in particular M 28, M 62, Ter1, Ter 5 and Ter 9, which also have considerably higher BD capture probabilities for similar $f_\\mathrm{bd}$. {However, these clusters are also relatively close to the galactic centre with low galactic latitude. Their high densities, large distances and possible extinction make these targets challenging for future transit surveys. A number of clusters (e.g. NGC 6656, NGC 6752 and $\\omega$ Cen) are relatively nearby and have similar $P_\\mathrm{capt}$ to 47 Tuc, thus representing promising targets for future transit surveys. However, overall, 47 Tuc remains possibly the best target given it has a large number of high mass stars that were not surveyed by \\citetalias{Gil00} and has the highest estimated $P_\\mathrm{capt}$ of the globular clusters that are not in the galactic centre. }}\n\n\\begin{table*}\n\\centering \n \\begin{tabular}{c c c c c c c c c} \n \\hline\nCluster & $d$ [kpc] & $l$ [$^\\circ$] & $b$ [$^\\circ$] & $\\log \\, N_{*,0.52-0.88}$ & $\\log n_{*,\\mathrm{hm}}$ [pc$^{-3}$] & $\\sigma_{v,\\mathrm{hm}}$ [km\/s] & $\\log [P_\\mathrm{capt}(0.7 \\, M_\\odot)\/f_\\mathrm{bd}]$ \\\\\n\\hline\n\\rowcolor{Gray}\nNGC 104 (47 Tuc) & $4.52 $& $305.89$ & $-44.89 $& $5.73$ & $2.35$ & $3.65 $ & $-3.38$ \\\\\nNGC 3201 & $4.74 $& $277.23$ & $8.64 $& $4.98$ & $1.49$ & $2.83 $ & $-4.09$ \\\\\n\\rowcolor{Gray}\nNGC 4372 & $5.71 $& $300.99$ & $-9.88 $& $5.07$ & $1.28$ & $2.75 $ & $-4.28$ \\\\\nNGC 4833 & $6.48 $& $303.60$ & $-8.02 $& $5.09$ & $2.06$ & $3.26 $ & $-3.60$ \\\\\n\\rowcolor{Gray}\nNGC 5139 ($\\omega$ Cen) & $5.43 $& $309.10$ & $14.97 $& $6.34$ & $2.30$ & $3.84 $ & $-3.46$ \\\\\nNGC 6218 & $5.11 $& $15.72$ & $26.31 $& $4.80$ & $1.99$ & $3.11 $ & $-3.64$ \\\\\n\\rowcolor{Gray}\nNGC 6254 (M 10) & $5.07 $& $15.14$ & $23.08 $& $5.09$ & $2.04$ & $3.24 $ & $-3.62$ \\\\\nNGC 6266 (M 62) & $6.41 $& $353.57$ & $7.32 $& $5.56$ & $3.41$ & $4.29 $ & $-2.41$ \\\\\n\\rowcolor{Gray}\nNGC 6304 & $6.15 $& $355.83$ & $5.38 $& $4.87$ & $2.00$ & $3.13 $ & $-3.64$ \\\\\nTer 1 & $5.67 $& $357.56$ & $0.99 $& $4.95$ & $2.97$ & $3.80 $ & $-2.78$ \\\\\n\\rowcolor{Gray}\nTer 5 & $6.62 $& $3.84$ & $1.69 $& $5.75$ & $3.04$ & $4.11 $ & $-2.76$ \\\\\nTer 9 & $5.77 $& $3.60$ & $-1.99 $& $4.85$ & $3.02$ & $3.80 $ & $-2.73$ \\\\\n\\rowcolor{Gray}\nNGC 6553 & $5.33 $& $5.25$ & $-3.02 $& $5.23$ & $2.26$ & $3.43 $ & $-3.43$ \\\\\nNGC 6626 (M 28) & $5.37 $& $7.80$ & $-5.58 $& $5.25$ & $3.19$ & $4.05 $ & $-2.60$ \\\\\n\\rowcolor{Gray}\nNGC 6656 (M 22) & $3.30 $& $9.89$ & $-7.55 $& $5.45$ & $2.29$ & $3.52 $ & $-3.42$ \\\\\nNGC 6752 & $4.12 $& $336.49$ & $-25.63 $& $5.22$ & $2.06$ & $3.28 $ & $-3.60$ \\\\\n\\rowcolor{Gray}\nNGC 6809 (M 55) & $5.35 $& $8.79$ & $-23.27 $& $5.06$ & $1.54$ & $2.91 $ & $-4.06$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Local globular cluster parameters from the N-body models of \\citet{Hilker20}, selected to be closer than $7$~kpc and more massive than $10^5\\, M_\\odot$. {The number of stars $N_{*, 0.52-0.88}$ is estimated by dividing the total mass by $0.5\\,M_\\odot$ then multiplying by $0.297$, the approximate fraction of stars with masses $0.52 - 0.88\\,M_\\odot$.} The stellar number density, $n_*$, and (one dimensional) velocity dispersion, $\\sigma_v$ are taken inside the half-mass radius. The last column is obtained by scaling the results for 47 Tuc using equation~\\ref{eq:GC_scaled}. }\n\\label{table:GC_scaled}\n\\end{table*}\n\n\n\n\\subsection{Caveats for capture rates}\n\\label{sec:caveats}\n\n{We have explored BD capture rates in detail and suggested that sufficiently large transit surveys can put upper limits on BD formation rates. However, it is possible that the present day short period companion rates are influenced by other physical mechanisms. Factors that may alter the rates of short period BD companions include (although not necessarily limited to): }\n\\begin{itemize}\n \\item \\textit{Primordial mass segregation}: We have demonstrated that once a cluster become mass segregated, BD tidal capture becomes inefficient. If a population is primordially segregated, this would similarly reduce the capture efficiency.\n \\item \\textit{Time-scale for circularisation:} In Paper II we explore the time-scale on which a migrating planet may undergo a dynamical perturbation while circularising. Following \\citetalias{Bon03}, we have assumed that this time-scale is short for a tidally captured BD \\citep{Mardling96}. However, if this is not the case then perturbations after the initial tidal encounter may curtail tidal circularisation and therefore prevent the formation of the {tight BD-star binary that can be detected through transit}. \n \\item \\textit{Tidal inspiral of BDs:} {Evidence for the correlation of hot Jupiter occurrence with cold stellar kinematics may originate from the inspiral of close companions onto the central star on Gyr timescales \\citep[][]{Hamer19}. If close sub-stellar companions do inspiral on these time-scales, then a similar process {may operate on} tidally captured BDs. However, hot Jupiters appear to be retained in the dense cluster M67 \\citep[][]{Brucalassi16} which has an age of $\\sim 4.5$~Gyr, such that this would require a relatively narrow range of inspiral timescales \\citep[see also discussion in Section 4.3 of][]{Winter21}.}\n \\item \\textit{Evacuation of BDs:} Apart from mass segregation due to two-body relaxation, low mass stars and BDs can be further evacuated from the central regions of the globular cluster by alternative heating mechanisms. For example, black hole subsystems may induce dynamically heating and eject low mass objects such as BDs to the cluster halo \\citep{Breen13,Giersz19}. However, this process occurs on a time-scale longer than the half-mass relaxation time-scale ($\\sim 3$~Gyr for 47 Tuc). The time-scale on which the majority of BD captures occurs in our models is $\\lesssim 2$~Gyr, while segregation on longer time-scales may not strongly influence capture rates. Similarly, heating due to tidal shocks during to passages through the galactic plane may operate time-scales comparable to two-body relaxation \\citep{Gnedin99}. If these mechanisms significantly reduce the stellar density after $\\sim 12$~Gyr, this would also suggest a moderately higher initial stellar density required to reproduce the present day density profile. These considerations may therefore increase the initial capture rate and subsequently reduce it due to enhanced mass segregation. We do not explore these possibilities quantitatively in this work. \n\\end{itemize}\n\n{An absence of close sub-stellar companions would therefore suggest either that BD formation is suppressed or that one of the above processes (or unconsidered alternative) is operating. In any case, non-detection in an increased sample of stars would require explanation and future survey campaigns are therefore merited.}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this work, we have explored the apparent absence of close-in sub-stellar companions in the globular cluster 47 Tuc from a theoretical perspective. We applied a Monte Carlo model using the \\textsc{Mocca} code \\citep{Hyp13, Giersz13} for the dynamical evolution of the globular cluster. Using this model, we compute the rates of tidal BD capture over its lifetime.\n\nOur results indicate lower capture efficiency than previous estimates \\citep{Bon03}. The reasons for this are subtle, but fundamentally originate from the rapid decrease of the tidal capture cross section with decreasing stellar mass. This is particularly true for environments with velocity dispersions as high as globular clusters. Once mass segregation operates, BDs and low mass stars are preferentially found in the same spatial location. Therefore the global tidal capture efficiency drops precipitously, such that the current constraints {cannot rule out that the frequency of BDs in the IMF is as high in 47 Tuc as in the galactic field.}\n\nThese considerations also lead to a steep scaling of the capture probability with stellar mass. For initial number of BDs $N_\\mathrm{bd}$ and stars $N_*$, those stars that have not reached the end of their main sequence have a lifetime capture probability:\n\\begin{equation}\n P_\\mathrm{capt} = 1.1 \\times 10^{-3} \\frac{N_\\mathrm{bd}}{N_*}\\cdot \\left(\\frac{m_*}{1\\, M_\\odot} \\right)^{2.7}.\n\\end{equation}The large exponent means that any constraints on the initial BD ratio are strongly dependent on the mass function of stars that are surveyed for close companions. For the typical masses of the stars surveyed by \\citet[][$0.52 \\,M_\\odot \\lesssim m_* \\lesssim 0.88\\, M_\\odot$]{Gil00} and equal numbers of BDs and stars, this yields capture probabilities that are comparable to the upper limit constraint on close sub-stellar companions ($P_\\mathrm{capt}\\lesssim 4\\cdot 10^{-4}$). \n\n\n\nFinally we conclude that, while the current constraints on the frequency of close sub-stellar companions {cannot rule out that the incidence of BDs in 47 Tuc is as high as it is in the field,} stronger constraints can be obtained by surveying a larger number of relatively high mass stars. {Such an exercise may also be achieved aggregating across several globular clusters.} We therefore estimate the capture rates in local globular clusters for a similar mass range of stars to those surveyed in 47 Tuc. The estimated capture rates are summarised in Table~\\ref{table:GC_scaled}. {We suggest that 47 Tuc remains among the most promising targets for follow up, with a convenient location and a large number of relatively high mass stars that have not yet been monitored for short period sub-stellar companions. A number of other globular clusters, such as $\\omega$ Cen, may also represent feasible targets for transit surveys. Our results offer motivation and interpretation for future transit surveys of globular clusters.}\n\n\n\\section*{Acknowledgements}\n\nWe thank the anonymous referee for their careful reading that helped clarify the manuscript and Abbas Askar for his helpful comments on dynamical models for 47 Tucanae. AJW acknowledges funding from an Alexander von Humboldt Stiftung Postdoctoral Research Fellowship. GR acknowledges support from the Netherlands Organisation for Scientific Research (NWO, program number 016.Veni.192.233) and from an STFC Ernest Rutherford Fellowship (grant number ST\/T003855\/1). This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 681601) and been supported by the DISCSIM project, grant agreement 341137 funded by the ERC under ERC-2013-ADG. \n\n\\section*{Data availability}\n\nAll data in this article is available from the corresponding author upon reasonable request.\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn this paper, we study the structure of parabolic induction of admissible representations of reductive groups defined over a non-archimedean local field $F$ of characteristic 0. Of course, the problem goes back to the era of Harish-Chandra and marches on in the century of Langlands for its deep connection with number theory, many great mathematicians have devoted their efforts on this topic (cf. \\cite{knappstein,knapp1976classification,langlands,jacquet,howe1976any,bernstein1977induced,speh1980reducibility,kazhdan1987proof,casselman1995introduction,shahidi1990proof,moeglin2002construction,moeglin2003points}).\n\nLet $G$ be a connected reductive group defined over $F$, and $P=MN$ be a parabolic subgroup of $G$ with $M$ its Levi subgroup and $N$ nilpotent radical of $P$. For an irreducible admissible representation $(\\sigma, V_\\sigma)$ of $M$, a fundamental question originating from Harish-Chandra's theory is to give a reasonable criterion of the reducibility of the parabolic induction $I^G_P(\\sigma)$:\n\\[I^G_P(\\sigma)=Ind^G_P(\\sigma):=\\{f:G\\rightarrow V_\\sigma \\mbox{ smooth }|f(mng)=\\delta_P(m)^{\\frac{1}{2}}\\sigma(m)f(g), \\forall m\\in M,n\\in N,g\\in G \\}. \\]\nIf $G$ is a finite group, such a question is answered perfectly by Mackey's theory. Indeed, analogous Mackey theory does exist for our $G$ which is the so-called Bernstein--Zelevinsky geometrical lemma (cf. \\cite{bernstein1977induced,casselman1995introduction,waldspurger2003formule}), and analogous simple criterion does exist for tempered inductions $I^G_P(\\sigma)$, i.e. $\\sigma$ is a discrete series representation. Such a criterion is the so-called Knapp--Stein $R$-group theory (cf. \\cite{knappstein,knapp1975singular,silberger1978knapp,luo2017R}). But there is an essential obstruction for general $\\sigma$: the restriction functor, i.e. Jacquet functor does not preserve unitarity as opposed to the finite group case. If $\\sigma$ is a supercuspidal representation, i.e. does not arise from proper parabolic inductions, a simple criterion has been discovered recently which originates from Muller's criterion for principal series, i.e. $\\sigma$ is a character (cf. \\cite{muller1979integrales,kato1982irreducible,luo2018muller}). For general $\\sigma$ and general $G$, there is no essential progress (to my best knowledge). Inspired by our Muller type irreducibility criterion for $\\sigma$ supercuspidal, we give a first structural answer for a large class of $\\sigma$. Denote by $\\rho$ the supercuspidal support of $\\sigma$ on $P_0=M_0N_0\\subset P=MN$, i.e. $\\sigma\\in JH(I^M_{M\\cap P_0}(\\rho))$ the set of Jordan--H\\\"{o}lder constituents of $I^M_{M\\cap P_0}(\\rho)$, then our universal hierarchical irreducibility criterion is as follows: please see the context for the detail,\n\\begin{thm}(cf. Theorem \\ref{prod}) Assume the reducibility conditions given by Muller type irreducibility criterion for $I^G_{P_0}(\\rho)$ lie in $M$, then\n\t$I^G_P(\\tau)$ is always irreducible for any $\\tau\\in JH(I^M_{M\\cap P_0}(\\rho))$.\n\\end{thm} \nSuch a criterion exists for tempered inductions which is a key ingredient of proving Howe's finiteness conjecture (cf. \\cite{clozel1989orbital,luo2017howe}). Indeed, Clozel also proposed a beautiful conjecture as a key ingredient in his first attempt to prove Howe's finiteness conjecture (see \\cite{clozel1985conjecture}). As an application of the above theorem, we serve you a simple intuitive proof of such a beautiful conjecture under the assumption that it holds for co-rank one cases.\n\nLet us take a brief look at the main idea of the proof of Theorem \\ref{prod} for regular $\\rho$, i.e. no non-trivial Weyl element fixes it. Our argument is quite classical. It relies heavily on the Bernstein--Zelevinsky geometrical lemma and the exactness of Jacquet functor (cf. \\cite{bernstein1977induced,casselman1995introduction,silberger2015introduction,waldspurger2003formule}): the constituents of the generalized principal series $I^G_{P_0}(\\rho)$ are parametrized by the partition of the set of the relative Weyl group $W_{M_0}:=N_G(M_0)\/M_0$. If our reducibility conditions of $I^G_{P_0}(\\rho)$ given by our Muller type irreducibility criterion lie in some standard Levi subgroup $M$, then it gives rise to a partition of $W_{M_0}^M:=N_M(M_0)\/M_0\\subset W_{M_0}$. Then to show the irreducibility of $I^G_P(\\sigma)$ for any $\\sigma\\in JH(I^M_{M\\cap P_0}(\\rho))$, the novelty of our argument is to find a good set $S$ of representatives of the quotient $W_{M_0}^M\\backslash W_{M_0}$, such that \n\\[I^G_P(\\rho^{w_1})\\simeq I^G_P(\\rho^{w_1w}) \\]\nfor any $w_1\\in W^M_{M_0}$ and $w\\in S$.\n\nIt seems that our argument is quite restrictive. But it also seems that this may be the only universal argument we could come up with nowadays which has been practiced intelligently by Bernstein--Zelevinsky, Silberger, Jantzen, Moeglin, the Tadi{\\'c} school etc (please refer to \\cite{bernstein1977induced,jantzen1997supports,moeglin2002construction,tadic1998regular,lapid2017some} for a glimpse). Inspired by a conjectural Muller type irreducibility criterion for general parabolic inductions (see \\cite[Conjecture 4.4]{luo2018muller}), a detailed study of the structure of Jacquet modules of a general parabolic induction may give us hope to prove the following conjectural universal irreducibility structure (Please see the context for the details):\n\\begin{conj}(see Conjecture \\ref{conj})\n\tGiven a parabolic induction $I^G_{P=MN}(\\sigma)$ with $\\sigma$ an irreducible admissible representation of $M$, if the ``reducibility conditions'' of $I^G_P(\\sigma)$ lie in some standard Levi subgroup $L$ of a parabolic subgroup $Q=LV\\subset P=MN$, then $I^G_Q(\\tau)$ is always irreducible for any $\\tau\\in JH(I^L_{L\\cap P}(\\sigma))$.\n\\end{conj}\n\\begin{rem}\n\tAt first glance, Conjecture \\ref{conj} may look meaningless and ridiculous. But some supportive examples of a conjectural Muller type irreducibility criterion for general parabolic induction, i.e. \\cite[Conjecture 4.4]{luo2018muller} could guide us to a right direction. Those examples are parabolic induction representations inducing from essentially discrete series and standard modules which many mathematicians have investigated since the era of Harish-Chandra.\n\\end{rem}\nInspired by Goldberg and Jantzen's product formulas for quasi-split classical groups (cf. \\cite{goldberg1994reducibility,jantzen1997supports,jantzen2005duality}), another direction we could cook up with the help of our new notion of $R$-group is a Goldberg--Jantzen type product formula as follows: (Please see the context for the details)\n\\begin{conj}(see Conjecture \\ref{gjconj})\n\tLet $\\nu\\in \\mathfrak{a}_M^*$ and $\\sigma$ be a unitary supercuspidal representation of $M$. For the generalized principal series representation $I^G_P(\\nu,\\sigma):=Ind^G_P(\\sigma\\otimes\\nu)$ of $G$, there is a one-one correspondence:\n\t\\[JH(I^G_P(\\nu,\\sigma))\\leftrightarrows \\prod_i JH(I^{M_i}_{M_i\\cap P}(\\nu,\\sigma)). \\]\n\\end{conj}\nHere $\\Phi_\\sigma:=\\{\\alpha\\in\\Phi_M:~w_\\alpha.\\sigma=\\sigma \\}$ is a root system and decomposes into irreducible pieces, i.e. $\\Phi_\\sigma=\\sqcup_i\\Phi_{\\sigma,i}$. Each irreducible root system $\\Phi_{\\sigma,i}$ gives rise to a Levi subgroup $M_i\\supset M$ which is defined by\n\\[M_i:=C_G((\\bigcap\\limits_{\\alpha\\in\\Phi_{\\sigma,i}}Ker(\\alpha) )^0). \\]\nIn the meantime we assume that $R_\\sigma$ associated to $\\sigma$ decomposes into a product of small pieces in the same pattern as does $\\Phi_\\sigma$, i.e.\n\\[R_\\sigma=\\prod_i R_{\\sigma,i} \\]\nwith $R_{\\sigma,i}$ a subgroup of the relative Weyl group $W^{M_i}_M=N_{M_i}(M)\/M$ of $M$ in $M_i$.\n\nAs a by-product of Theorem \\ref{prod}, we could reduce Clozel's finiteness conjecture of special exponents to co-rank one cases. There is no harm to assume that $G$ is of compact center. For a discrete series representation $\\sigma$ of $G$, we denote its associated supercuspidal support to be $\\rho$ which is a supercuspidal representation of some Levi subgroup $M$ of $G$. Denote by $\\omega_\\rho$ the unramified part of the central character of $\\rho$, i.e. $\\omega_\\rho\\in \\mathfrak{a}_{M,\\mathbb{C}}^\\star:=Hom_F(M,\\mathbb{G}_m)\\otimes \\mathbb{C}$. Such a character is called a \\underline{special exponent}. Clozel conjectured that the set of special exponents is finite (see \\cite{clozel1985conjecture,clozel1989orbital}).\n\n\\begin{thm}(cf. Theorem \\ref{clozel})\n\tThe set of special exponents is finite provided it is true for co-rank one cases.\n\\end{thm}\nIndeed, the generic co-rank one case is a result of the profound Langlands--Shahidi theory, and the general co-rank one case follows from two conjectures of Shahidi (cf. \\cite[Conjectures 9.2 \\& 9.4]{shahidi1990proof}). Those two conjectures for classical groups in some sense are by-products of Arthur's standard model argument in his monumental book \\cite{arthur2013endoscopic}. The main idea of the proof of Clozel's finiteness conjecture is as follows:\n\\begin{enumerate}[(i)]\n\t\\item A full induced representation $I^G_P(\\sigma)$ can never be discrete series.\n\t\\item An invertible matrix has only one solution.\n\\end{enumerate}\nTo illustrate the simple ideas, let us take a look at the real parts of special exponents, if $I^G_P(\\rho)$ contains a discrete series subquotient, then the reducibility conditions must generate the whole vector space $\\mathfrak{a}^\\star_{M,\\mathbb{C}}$. Otherwise, there exists a proper parabolic subgroup $Q=LV\\supset P$ such that the reducibility conditions lie in $L$, then our universal hierarchical irreducibility criterion implies that $I^G_Q(\\tau)$ is always irreducible for any $\\tau\\in JH(I^L_{L\\cap P}(\\rho))$, whence any constituent of $I^G_P(\\rho)$ can never be discrete series. Contradiction. Thus $\\mathfrak{a}^\\star_{M,\\mathbb{C}}$ is generated by the reducibility conditions. Indeed, we recently learned that such a claim is also a corollary of an old result of Harish-Chandra (cf. \\cite[Theorem 5.4.5.7]{silberger2015introduction}, \\cite[Theorem 3.9.1]{silberger1981discrete} or \\cite[Corollary 8.7]{heiermann2004decomposition}). Under the assumption that the set of special exponents is finite for co-rank one cases, those reducibility conditions form a finite set of hyperplanes in $\\mathfrak{a}^\\star_{M,\\mathbb{C}}$, therefore the finiteness for high rank cases follows easily from the fact that there exists only one solution for an invertible matrix. At last, we would like to mention that Clozel proposed an analogous strong global conjecture of finiteness of poles of Eisenstein series constructed from cusp forms which would simplify Arthur's trace formula machine significantly. We hope that the simple intuitive proof of Clozel's local finiteness conjecture could shed some light on his global conjecture in our future work.\n\nWe end the introduction by recalling briefly the structure of the paper. In Section 2, we prepare some necessary notation. In Section 3, we state and prove the universal hierarchical irreducibility criterion for Harish-Chandra parabolic inductions. In the end, as its first application, a new understanding of the generic irreducibility property of parabolic inductions is provided. In Section 4, with the aide of the universal hierarchical irreducibility criterion in the previous section, a simple intuitive proof of Clozel's finiteness conjecture is served under the assumption that it holds for the co-rank one case. \n\n\\section{Preliminaries}\nLet $G$ be a connected reductive group defined over a non-archimedean local field $F$ of characteristic 0. Denote by $|-|_F$ the absolute value, by $\\mathfrak{w}$ the uniformizer and by $q$ the cardinality of the residue field of $F$. Fix a minimal parabolic subgroup $B=TU$ of $G$ with $T$ a minimal Levi subgroup and $U$ a maximal unipotent subgroup of $G$, and let $P=MN\\supset B=TU$ be a standard parabolic subgroup of $G$ with $M$ the Levi subgroup and $N$ the unipotent radical.\n\n\\subsection{Structure theory}Let $X(M)_F$ be the group of $F$-rational characters of $M$, and set \n\\[\\mathfrak{a}_M=Hom(X(M)_F,\\mathbb{R}),\\qquad\\mathfrak{a}^\\star_{M,\\mathbb{C}}=\\mathfrak{a}^\\star_M\\otimes_\\mathbb{R} \\mathbb{C}, \\]\nwhere\n\\[\\mathfrak{a}^\\star_M=X(M)_F\\otimes_\\mathbb{Z}\\mathbb{R} \\]\ndenotes the dual of $\\mathfrak{a}_M$. Recall that the Harish-Chandra homomorphism $H_P:M\\longrightarrow\\mathfrak{a}_M$ is defined by\n\\[q^{\\left< \\chi,H_P(m)\\right>}=|\\chi(m)|_F \\] \nfor all $\\chi\\in X(M)_F$.\n\nNext, let $\\Phi$ be the root system of $G$ with respect to $T$, and $\\Delta$ be the set of simple roots determined by $U$. For $\\alpha\\in \\Phi$, we denote by $\\alpha^\\vee$ the associated coroot, and by $w_\\alpha$ the associated reflection in the Weyl group $W=W^G$ of $T$ in $G$ with\n\\[W:=N_G(T)\/T=\\left. \\]\nDenote by $w_0^G$ the longest Weyl element in $W$, and similarly by $w_0^M$ the longest Weyl element in the Weyl group $W^M:=N_M(T)\/T$ of a Levi subgroup $M$. \n\nLikewise, we denote by $\\Phi_M$ (resp. $\\Phi_M^L$) the reduced relative root system of $M$ in $G$ (resp. the Levi subgroup $M\\subset L$), by $\\Delta_M$ the set of relative simple roots determined by $N$ and by $W_M:=N_G(M)\/M$ (resp. $W_M^L$) the relative Weyl group of $M$ in $G$ (resp. $L$). In general, a relative reflection $\\omega_\\alpha:=w_0^{M_\\alpha}w_0^M$ with respect to a relative root $\\alpha$ does not preserve our Levi subgroup $M$. Denote by $\\Phi^0_M$ (resp. $\\Phi^{L,0}_M$) the set of those relative roots which contribute reflections in $W_M$ (resp. $W_M^L$). It is easy to see that $W_M$ preserves $\\Phi_M$, and further $\\Phi_M^0$ as well, as $\\omega_{w.\\alpha}=w\\omega_\\alpha w^{-1}$. Note that $W_M$ (resp. $W_M^{L,0}$) in general is larger than $W_M^0$ (resp. $W_M^{L,0}$) the one generated by those relative reflections in $G$ (resp. $L$). Denote by $\\Phi_M(P)$ the set of reduced roots of $M$ in $P$.\n\nRecall that the canonical pairing $$\\left<-,-\\right>:~\\mathfrak{a}^\\star_M\\times \\mathfrak{a}_M\\longrightarrow\\mathbb{R}$$ suggests that each $\\alpha\\in \\Phi_M$ will enjoy a one parameter subgroup $H_{\\alpha^\\vee}(F^\\times)$ of $M$ satisfying: for $x\\in F^\\times$ and $\\beta\\in \\mathfrak{a}^\\star_M$,\n\\[\\beta(H_{\\alpha^\\vee}(x))=x^{\\left<\\beta,\\alpha^\\vee\\right>}. \\]\n\n\\subsection{Parabolic induction and Jacquet module}For $P=MN$ a parabolic subgroup of $G$ and an admissible representation $(\\sigma,V_\\sigma)~ (resp.~(\\pi,V_\\pi))$ of $M~(resp.~G)$, we have the following normalized parabolic induction of $P$ to $G$ which is a representation of $G$\n\\[I_P^G(\\sigma)=Ind_P^G(\\sigma):=\\{\\mbox{smooth }f:G\\rightarrow V_\\sigma|~f(nmg)=\\delta_P(m)^{1\/2}\\sigma(m)f(g), \\forall n\\in N, m\\in M~and~g\\in G\\} \\]\nwith $\\delta_P$ stands for the modulus character of $P$, i.e., denote by $\\mathfrak{n}$ the Lie algebra of $N$,\n\\[\\delta_P(nm)=|det~Ad_\\mathfrak{n}(m)|_F, \\]\nand the normalized Jacquet module $J_M(\\pi)$ with respect to $P$ which is a representation of $M$\n\\[\\pi_N:=V_\\pi\/\\left<\\pi(n)e-e:~n\\in N,e\\in V_\\pi\\right>. \\] \nGiven an irreducible admissible representation $\\sigma$ of $M$ and $\\nu\\in \\mathfrak{a}^\\star_{M}$, let $I(\\nu,\\sigma)$ be the representation of $G$ induced from $\\sigma$ and $\\nu$ as follows:\n\\[I(\\nu,\\sigma)=Ind_P^G(\\sigma\\otimes q^{\\left<\\nu,H_P(-)\\right>}) .\\]\nWe denote by $JH(I_P^G(\\sigma))$ the set of Jordan--H\\\"{o}lder constituents of the parabolic induction $I^G_P(\\sigma)$, and define the action of $w\\in W_M$ on representations $\\sigma$ of $M$ to be $w.\\sigma=\\sigma\\circ Ad(w)^{-1}$ and $\\sigma^w=\\sigma\\circ Ad(w)$.\n\n\\subsection{$R$-group}\nIn \\cite{muller1979integrales}, for a principal series $I(\\lambda)$ of $G$, she defines a subgroup $W_\\lambda^1$ of the Weyl group $W$ governing the reducibility of the ``unitary'' part of principal series on the Levi level, which is indeed the Knapp--Stein $R$-group as follows (cf. \\cite{winarsky1978reducibility,keys1982decomposition}), \n\\begin{align*}\n\\Phi_{\\lambda}^0&:=\\{\\alpha\\in \\Phi:~\\lambda_\\alpha=Id \\},\\\\\nW_{\\lambda}^0&:=\\left,\\\\\nW^1_\\lambda&:=\\{w\\in W_\\lambda:~w.(\\Phi_\\lambda^0)^+>0 \\},\\\\\nW_\\lambda&:=\\{w\\in W:~w.\\lambda=\\lambda \\}.\n\\end{align*} \nIn view of \\cite[Lemma I.1.8]{waldspurger2003formule}, one has\n\\[W_\\lambda=W_\\lambda^0\\rtimes W_\\lambda^1. \\]\nFollowing the Knapp--Stein R-group theory (cf. \\cite{silberger2015introduction}), we denote by $R_\\lambda$ the subgroup $W_\\lambda^1$.\n\nLikewise, for generalized principal series $I^G_P(\\sigma)$ (cf. \\cite{luo2018muller}),\n\\begin{align*}\n\\Phi_{\\sigma}^0&:=\\{\\alpha\\in \\Phi_M^0:~w_\\alpha.\\sigma=\\sigma \\},\\\\\nW_{\\sigma}^0&:=\\left,\\\\\nW^1_{\\sigma}&:=\\{w\\in W_{\\sigma}:~w.(\\Phi_{\\sigma}^0)^+>0 \\},\\\\\nW_{\\sigma}&:=\\{w\\in W_M:~w.\\sigma=\\sigma \\}.\n\\end{align*}\nVia \\cite[Lemma I.1.8]{waldspurger2003formule}, we have\n\\[W_{\\sigma}=W_{\\sigma}^0\\rtimes W_{\\sigma}^1, \\]\nand we denote $R_{\\sigma}$ to be $W_{\\sigma}^1$ following tradition, but it is not the exact Knapp--Stein $R$-group in the sense of Silberger.\n\n\\subsection{Special exponent} There is no harm to assume that $G$ is of compact center. For a discrete series representation $\\pi$ of $G$, we denote its associated supercuspidal support to be $\\sigma$ which is a supercuspidal representation of some Levi subgroup $M$ of $G$. Denote by $\\omega_\\sigma$ the unramified part of the central character of $\\sigma$, i.e. $\\omega_\\sigma\\in \\mathfrak{a}_{M,\\mathbb{C}}^\\star$. Such a character is called a \\underline{special exponent}.\n\n\\section{Universal Hierarchical Structure of Reducibility}\n\\subsection{A product formula}\nIn this subsection, we prove a key observation of the decomposition of parabolic induction which opens a gate to understand some of the classical results\/conjectures, for example the generic irreducibility property of parabolic induction and Clozel's finiteness conjecture of special exponents.\n\nRecall that $G$ is a connected reductive group defined over $F$ with the set of simple roots $\\Delta$, $P=MN$ is a standard parabolic subgroup of $G$ associated to $\\Theta_M\\subset \\Delta$ and $\\sigma$ is a supercuspidal representation of $M$ (not necessary unitary), one forms a parabolic induction $I_P^G(\\sigma)$. Then our ``product formula'' is designed to ask the following question\n\\[\\mbox{When does the reducibility of $I^G_P(\\sigma)$ only happen on the Levi-level?\\tag*{$(\\star)$}} \\]\ni.e.\n\\[\\mbox{What is a reasonble condition for the irreducibility of $I^G_{Q=LV}(\\tau)$ for all $\\tau\\in JH(I_{L\\cap P}^L(\\sigma))$?} \\]\nThe answer traces back to a beautiful theorem of I. Muller \\cite{muller1979integrales,kato1982irreducible} which provides a natural criterion of the irreducibility of principal series, and its generalized version for generalized principal series \\cite{luo2018muller} using the Knapp--Stein $R$-group and co-rank one reducibility. As the irreducibility is governed by the Knapp--Stein $R$-group and the co-rank one reducibility, a natural candidate for ($\\star$) is to assume that those governing conditions occur only on the Levi-level. To be more precise, let $Q=LV$ be a standard parabolic subgroup associated to $\\Theta_L$ with $\\Theta_M\\subset\\Theta_L\\subset \\Delta$, then our working assumption for $(\\star)$ is as follows.\n\n{\\bf \\underline{Working Hypothesis}}:\n\\leavevmode\n\\begin{enumerate}[(i)]\n\t\\item (\\underline{Rank-one reducibility}) The co-rank one reducibility only occurs within $L$, i.e.\n\t\\[I_M^{M_\\alpha}(\\sigma) \\mbox{ is reducible only for some }\\alpha\\in \\Phi_M^{L},~i.e.~\\alpha\\in \\Phi_M^{L,0}\\mbox{ (cf. \\cite[Theorem 7.1.4]{casselman1995introduction})}. \\] \n\t\\item (\\underline{$R$-group}) The $R$-group $R_\\sigma$ associated to $\\sigma$ is a subgroup of $W^L_M$, i.e.\n\t\\[R_\\sigma\\subset W^L_M. \\]\n\\end{enumerate}\nUnder those hypothesis, via the Jacquet module machine, the confirmation of $(\\star)$ results from the associativity property of intertwining operators and the following observation\/fact (cf. \\cite{bernstein1977induced,casselman1995introduction,moeglin_waldspurger_1995,waldspurger2003formule}).\n\\begin{enumerate}[(i)]\n\t\\item (Bernstein--Zelevinsky geometrical lemma)\n\t\\[J_M( I_P^G(\\sigma))=\\sum\\limits_{w\\in W_M}\\sigma^w. \\]\n\t\\item (Bruhat--Tits decomposition)\n\t\\[W_M=W_M(L)W^L_M, \\tag{$\\star\\star$}\\]\n\twhere $W_M(L)$ is defined as follows:\n\t\\[W_M(L):=\\{w\\in W_M:~w.(\\Phi_M^{L,0})^+>0 \\}\/W_M^{L,1}, \\]\n\there $W_M^{L,1}$ is defined as in \\cite[Lemma 2.1]{luo2018muller}), i.e.\n\t\\[W_M^{L,1}:=\\{w\\in W_M^L:w.(\\Phi_M^{L,0})^+>0 \\}. \\]\n\\end{enumerate}\nTo be precise,\n\\begin{thm}(Universal hierarchical irreducibility criterion)\\label{prod}\n\tKeep the notions as above. Under the above {\\bf Working Hypothesis}, we have the following cardinality equality\n\t\\[\\#JH(I^G_P(\\sigma))=\\#JH(I_{L\\cap P}^L(\\sigma)). \\]\n\tIndeed, this is equivalent to saying that \n\t\\[I^G_Q(\\tau)\\mbox{ is always irreducible for any }\\tau \\in JH(I^L_{L\\cap P}(\\sigma)). \\]\n\\end{thm}\nBefore turning to the proof, let us first prove the Bruhat-Tits decomposition, i.e. $(\\star\\star)$, which is a generalization of \\cite[Lemma 1.1.2]{casselman1995introduction} for $M=T$, in what follows.\n\\begin{proof}[Proof of $(\\star\\star)$]\n\tFirst note that (cf. \\cite[Lemma 2.1]{luo2018muller}) \n\t\\[W_M:=W_M^0\\rtimes W_M^1 \\]\n\tand \n\t\\[W_M^L:=W_M^{L,0}\\rtimes W_M^{L,1}, \\]\n\twith\n\t\\[W_M^{1}:=\\{w\\in W_M:w.(\\Phi_M^{0})^+>0 \\}=\\{w\\in W_M:w.(\\Phi_M^{0})^+=(\\Phi_M^{0})^+ \\}. \\]\n\tSo \n\t\\[W_M\/W_M^L=\\{w\\in W_M:w.(\\Phi_M^{L,0})^+>0 \\}\/W_M^{L,1}. \\]\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{prod}]\n\tNote that the decomposition of $I^G_P(\\sigma)$ is a partition of $W_M$. Recall that $R_\\sigma$ is in general not the exact R-group in the sense of Knapp--Stein, as it is defined by\n\t\\[W_\\sigma=W_\\sigma^0\\rtimes R_\\sigma, \\]\n\twhere\n\t\\[W_\\sigma:=\\{w\\in W_M:~w.\\sigma=\\sigma \\},\\]\n\tand \n\t\\[W_\\sigma^0:=\\left. \\]\n\tBut reducibility coming from $W_\\sigma^0$ has been taken care of by the co-rank one reducibility condition which only occurs within $L$ by the assumption. On the other hand, the Knapp-- Stein $R$-group theory helps us control the multiplicity issue.\n\t\n\tTherefore it reduces to show that the non-zero intertwining operator $A(w,\\sigma)$ associated to $w\\in W_M(L)$ is an isomorphism, i.e.\n\t\\[A(w,\\sigma):~I_P^G(\\sigma)\\stackrel{\\sim}{\\longrightarrow} I_P^G(\\sigma^w). \\]\n\tRecall that $A(w,\\sigma)$ is defined as follows:\n\t\\[J_{P|P^w}(\\sigma^w)\\circ \\lambda(w):~I^G_P(\\sigma)\\longrightarrow I^G_{P^w}(\\sigma^w)\\longrightarrow I^G_P(\\sigma^w). \\]\n\tThus the above isomorphism claim follows from the associativity property of intertwining operators (cf.\\cite[IV.3 (4)]{waldspurger2003formule} or \\cite[Lemma 3.5]{luo2018R}), i.e.\n\t\\[J_{P|P^w}(\\sigma^w)J_{P^w|P}(\\sigma)=\\prod j_\\alpha(\\sigma)J_{P|P}(\\sigma), \\]\n\twhere $\\alpha$ runs over\n\t\\[\\Phi_M(P)\\bigcap \\Phi_M(\\overline{P^w}) \\]\n\twith $\\overline{P^w}$ the opposite parabolic subgroup of $P^w$, and $\\Phi_M(P)$ (resp. $\\Phi_M(\\overline{P^w})$) is the set of restricted roots of $M$ in $P$ (resp. $\\overline{P^w}$).\n\t\n\tNote that for $\\alpha\\in \\Phi_M(P)-\\Phi_M^0$, the associated co-rank one induction is always irreducible and $j_\\alpha(\\sigma)\\neq 0,\\infty$ (cf. \\cite[Corollary 1.8]{silberger1980special}). \n\t\n\tNote also that for $\\alpha\\in \\Phi_M^{0}-\\Phi_M^{L,0}$, the associated co-rank one induction is always irreducible by our Working Hypothesis, which implies that either $j_\\alpha(\\sigma)\\neq 0,\\infty$ for non-unitary induction (cf. \\cite[Corollary 1.8]{silberger1980special}), or $j_\\alpha(\\sigma)$ has a pole of order 2 (cf. \\cite[Proposition 2]{savin2007}). For the latter case, one can take the residue to get an isomorphism. \n\t\n\tTherefore one only needs to consider those $j_\\alpha(\\sigma)$ with $\\alpha\\in \\Phi_M^{L,0}$. \n\t\n\tFor those $\\alpha\\in \\Phi_M^{L,0}$, we have \n\t\\[w.(\\Phi_M^{L,0})^+>0, \\]\n\tso\n\t\\[\\Phi_M(P)\\bigcap \\Phi_M(\\overline{P^w})\\bigcap \\Phi_M^{L,0}=\\emptyset. \\]\n\tThus $A(w,\\sigma)$ is an isomorphism.\n\\end{proof}\nIt seems that our proof of Theorem \\ref{prod} is quite restrictive. But it also seems that this may be the only universal argument we could come up with nowadays which has since been practiced intelligently by Bernstein--Zelevinsky, Silberger, Jantzen, Moeglin, the Tadi{\\'c} school etc (please refer to \\cite{bernstein1977induced,jantzen1997supports,moeglin2002construction,tadic1998regular,lapid2017some} for a glimpse). Inspired by a conjectural Muller type irreducibility criterion for general parabolic inductions (see \\cite[Conjecture 4.4]{luo2018muller}), a detailed study of the structure of Jacquet modules of a general parabolic induction may give us hope to prove the following conjectural universal irreducibility structure:\n\\begin{conj}\\label{conj}\n\tGiven a parabolic induction $I^G_{P=MN}(\\sigma)$ with $\\sigma$ an irreducible admissible representation of $M$, if the reducibility conditions of $I^G_P(\\sigma)$ lie in some standard Levi subgroup $L$ of a parabolic subgroup $Q=LV\\subset P=MN$, then $I^G_Q(\\tau)$ is always irreducible for any $\\tau\\in JH(I^L_{L\\cap P}(\\sigma))$.\n\\end{conj}\n\\begin{rem}\n\tAt first glance, Conjecture \\ref{conj} may look meaningless and ridiculous. But some supportive examples of a conjectural Muller type irreducibility criterion for general parabolic induction, i.e. \\cite[Conjecture 4.4]{luo2018muller} could guide us to a right direction. Those examples are parabolic induction representations inducing from essentially discrete series and standard modules which many mathematicians have investigated since the era of Harish-Chandra.\n\\end{rem}\nDenote by $\\Theta_Q$ the associated subset of $\\Delta$ which determines the parabolic subgroup $Q=LV\\supset P=MN$ of $G$. Explicitly, we decompose $\\Theta_L=\\Theta_1\\sqcup\\cdots \\sqcup \\Theta_t $ into irreducible pieces, and accordingly $\\Theta_M=\\Theta_1^M\\sqcup \\cdots \\sqcup \\Theta_t^M$. Assume that $R_\\sigma$ decomposes into $R_\\sigma=R_1\\times \\cdots \\times R_t$ with respect to the decomposition of $\\Theta_L$, and a similar decomposition pattern holds for the co-rank one reducibility, i.e. co-rank one reducibility only occurs within $P_{\\Theta_i}=M_{\\Theta_i}N_{\\Theta_i}$ for $1\\leq i\\leq t$. Then we have \n\\begin{cor}[Product formula]\\label{prod1}\n\t\\[\\#(JH(I^G_P(\\sigma)))=\\prod_{i=1}^{t}\\#(JH(I_{M_{\\Theta_i^M}}^{M_{\\Theta_i}}(\\sigma))).\\]\n\\end{cor}\n\\begin{rem}\n\tVery recently, we learned that Jantzen has a beautiful product formula for split $Sp_{2n}$, $SO_{2n+1}$ and $O_{2n}$ which in some sense originates from Goldberg's product formula for tempered inductions (cf. \\cite{jantzen1997supports,jantzen2005duality,goldberg1994reducibility}). Note that our new $R$-group is always trivial for those groups in Goldberg and Jantzen's works. Inspired by their beautiful theorems, we would like to investigate what kind of general product formula we could prove under our new notion of $R$-group in our future work.\n\\end{rem}\nAs an instance, one version of Goldberg--Jantzen type product formula is as follows. Consider the tempered generalized principal series $I^G_P(\\sigma)$ with $\\sigma$ unitary supercuspidal representation of $M$, we know that $\\Phi_\\sigma:=\\{\\alpha\\in\\Phi_M:~w_\\alpha.\\sigma=\\sigma \\}$ is a root system, may be reducible. Decomposing $\\Phi_\\sigma$ into irreducible pieces, i.e. $\\Phi_\\sigma=\\sqcup_i\\Phi_{\\sigma,i}$. Each irreducible root system $\\Phi_{\\sigma,i}$ gives rise to a Levi subgroup $M_i\\supset M$ which is defined by\n\\[M_i:=C_G((\\bigcap\\limits_{\\alpha\\in\\Phi_{\\sigma,i}}Ker(\\alpha) )^0). \\]\nAssume that our new $R$-group $R_\\sigma$ associated to $\\sigma$ decomposes into a product of small pieces in the same pattern as does $\\Phi_\\sigma$, i.e.\n\\[R_\\sigma=\\prod_i R_{\\sigma,i} \\]\nwith $R_{\\sigma,i}$ a subgroup of the relative Weyl group $W^{M_i}_M=N_{M_i}(M)\/M$ of $M$ in $M_i$, then we have\n\\begin{conj}(Goldberg--Jantzen type product formula)\\label{gjconj}\n\tKeep the notions as above. For $\\nu\\in \\mathfrak{a}_M^*$, there is a one-one correspondence:\n\t\\[JH(I^G_P(\\nu,\\sigma))\\leftrightarrows \\prod_i JH(I^{M_i}_{M_i\\cap P}(\\nu,\\sigma)). \\]\n\\end{conj} \n\n\\subsection{Generic irreducibility property of parabolic induction}\nIn this subsection, we provide a new simple proof of the generic irreducibility property of parabolic inductions which plays an essential role in Harish-Chandra's Plancherel formula \\cite[IV.3]{waldspurger2003formule}.\n\nGiven an irreducible smooth representation $\\sigma$ of the Levi subgroup $M$ of a parabolic subgroup $P=MN$ in reductive group $G$, we form a family of normalized parabolic induction representations $I(\\nu,\\sigma)=Ind^G_P(\\sigma\\otimes \\nu)$ of $G$, where $\\nu$ varies in $\\mathfrak{a}_{M,\\mathbb{C}}^\\star$. The generic irreducibility property says that \n\\begin{thm}(Generic Irreducibility Theorem cf. \\cite{sauvageot})\n\tKeep the notions as above, we have\n\t\\[Irred_\\sigma:=\\{\\nu\\in \\mathfrak{a}_{M,\\mathbb{C}}^\\star:~I(\\nu,\\sigma) \\mbox{ is irreducible} \\}\\mbox{ is a non-trivial Zariski open subset.} \\]\n\\end{thm}\n\\begin{proof}\n\tIn view of the Langlands classification theorem, it reduces to the case where $\\pi$ is supercuspidal. Based on Theorem \\ref{prod} (or Muller type irreducibility criterion \\cite{luo2018muller}), the reducibility conditions are controlled by co-rank one reducibility and $R$-group. As there exists a unique reducibility point for the co-rank one case (cf. \\cite[Lemma 1.2 \\& 1.3]{silberger1980special}), thus it reduces to consider the $R$-group condition. Note that the $R$-group is a subgroup of $W_{\\sigma_\\nu}:=\\{w\\in W_M:~w.(\\sigma\\otimes\\nu)=\\sigma\\otimes \\nu \\}$, whence the non-trivial set $Irred_\\sigma$ is Zariski open. \n\\end{proof} \n\\section{Clozel's Finiteness Conjecture Of Special Exponents}\nIn this section, let us start with quoting Clozel's remark on Clozel's finiteness conjecture of special exponents in his second paper on Howe's finiteness conjecture \\cite[P. 3]{clozel1989orbital} as follows:\n\n\\textit{``We would like to finish the introduction with the remark that the stronger conjecture still retains some interest, although we do not know any obvious application. Here the analogy with the theory of automorphic forms is interesting. In the automorphic case, the analogue of the finiteness assumption about exponents would be the fact that the poles of Eisenstein series constructed from cusp forms on a given parabolic subgroup lie in a fixed, finite set independent of the inducing cusp form. This is a very strong conjecture, unknown even for $GL(n)$, although it would result from the conjectural description of the residues stated by Jacquet in [11]. If this conjecture was true, that would trivially imply that the operator defined by a smooth, $K$-finite function on the adelic group acting on the discrete spectrum is trace-class.''}\n\nAs an application of Theorem \\ref{prod}, under some conditions, we prove Clozel's finiteness conjecture of special exponents proposed in \\cite{clozel1985conjecture} which plays an essential role in Clozel's firt attempt to proving Howe's finiteness conjecture. Note that Clozel's finiteness conjecture may be checked directly for classical groups from Moeglin--Tadic's work on the classification of discrete series (cf. \\cite{moeglin2002construction,moeglin2007classification}). As the conjecture is much of a quantitative result, it should be proved with little forces, instead of resorting to such a big stick. Indeed, our proof is quite natural and may shed some lights on the global analogy. In what follows, we first recall some notions.\n\nThere is no harm to assume that $G$ is of compact center. Recall that for a discrete series representation $\\pi$ of $G$, we write its associated supercuspidal support as $\\sigma$ which is a supercuspidal representation of some Levi subgroup $M$ of $G$. Denote by $\\omega_\\sigma$ the unramified part of the central character of $\\sigma$, i.e. $\\omega_\\sigma\\in \\mathfrak{a}_{M,\\mathbb{C}}^\\star$. Such a character is called a \\underline{special exponent}.\n\\begin{thm}(Clozel's finiteness conjecture)\\label{clozel}\n\tThe set of special exponents is finite provided it holds for co-rank one cases.\n\\end{thm}\n\nBefore turning to the proof, let us first talk about the main idea.\n\nUnder the {\\bf Induction Assumption}, i.e.\n\\[\\mbox{Clozel's finiteness conjecture holds for the co-rank one case.}\\]\nOur proof of Clozel's finiteness conjecture for the general case rests on the following two novel observations:\n\\begin{enumerate}[(i)]\n\t\\item Theorem \\ref{prod}, or ``Product Formula''.\n\t\\item Irreducible induced representation can never be a discrete series.\n\\end{enumerate}\nRoughly speaking, with the help of Muller type theorem of generalized principal series, one knows that the decomposition of $I_P^G(\\sigma)$ is governed by the co-rank one reducibility and the $R$-group $R_\\sigma$. On the other hand, one knows that a full induced representation can not be a discrete series. In view of Theorem \\ref{prod}, thus in order to ensure $\\omega_\\sigma$ is a special exponent, those roots associated to the co-rank one reducibility and $R_\\sigma$ must generate the whole space $\\mathfrak{a}_{M,\\mathbb{C}}^\\star$. Then the conjecture follows from an easy fact of linear algebra, i.e. an invertible matrix has only one solution.\n\nTo be more precise, let $P_\\Theta=M_\\Theta N$ be a standard parabolic subroup of $G$ with $\\Theta\\subset \\Delta$, and let $\\sigma$ be a supercuspidal representation of $M_\\Theta$. Decomposing $\\Theta$ into irreducible pieces\n\\[\\Theta=\\Theta_1\\sqcup\\cdots\\sqcup\\Theta_n \\]\nAs $W_{M_\\Theta}$ acts on $M_\\Theta$, then it preserves the decomposition up to sign, so does $R_\\sigma$, i.e. preserving\n\\[\\pm\\Theta=\\pm\\Theta_1\\sqcup\\cdots\\sqcup\\pm\\Theta_n.\\] \nUnder the action of $R_\\sigma$ on $\\pm\\Theta$, we have a new decomposition of $\\pm\\Theta$ into irreducible pairs, i.e.\n\\[R_\\sigma\\hookrightarrow S_{\\pm n}, \\]\nwhere $S_{\\pm n}$ is the ``pseudo''-permutation group, i.e.\n\\[S_{\\pm n}:=\\{\\left( a_{i_1}\\cdots a_{i_k}\\right):~ a_{i_j}\\in \\{\\pm 1,\\cdots,\\pm n\\} \\}\/\\pm. \\]\nHere $\\pm $-equivalence means that\n\\[\\left(a_{i_1}\\cdots a_{i_k}\\right)=\\left((- a_{i_1})\\cdots (-a_{i_k})\\right) .\\]\nFor each simple permutation $s=\\left(a_{i_1}\\cdots a_{i_k}\\right)$, we define the associated roots as, up to scalar,\t\n\\[\\Phi_s:=\\{e_{i_j}-e_{i_l}:~1\\leq j