diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbwde" "b/data_all_eng_slimpj/shuffled/split2/finalzzbwde" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbwde" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe European Union (EU) is now composed from 27 countries \nand is considered as a major world leading power \\cite{eusite}.\nJanuary 2021 has seen Brexit officially taking place, triggering the \nwithdrawal of the United Kingdom (UK) from EU \\cite{wikibrexit}.\nThis event has important political, economical and social effects.\nHere we project and study its consequences from the view point\nof international trade between world countries.\nOur analysis is based on the UN COMTRADE database \\cite{comtrade} for\nthe multiproduct trade between world countries in \nrecent years. From this database we construct the world trade\nnetwork (WTN) and evaluate the influence and trade power \nof specific countries using the Google matrix analysis\nof the WTN. We consider 27 EU countries as a single\ntrade player having the trade exchange between EU and other countries.\nOur approach uses the Google matrix tools and algorithms developed for the WTN \n\\cite{wtn1,wtn2,wtn3,keu9} and other complex directed networks\n\\cite{rmp2015,politwiki}. The efficiency of the Google matrix and PageRank algorithms\nis well known from the World Wide Web network analysis \\cite{brin,meyer}.\n\nOur study shows that the Google matrix approach (GMA) allows to characterize \nin a more profound manner the trade power of countries \ncompared to the usual method relying on import and export analysis (IEA) between countries.\nGMA's deeper analysis power originates in the fact that it accounts for the multiplicity\nof transactions between countries while IEA only takes into account the \neffect of one step (direct link or relation) transactions. In this paper, we show that the \nworld trade network analysis with GMA identifies EU as the first trade player in the world, \nwell ahead of USA and China.\n\nThis paper is structured in the following way. Section~\\ref{sec:dataset} introduces first the UN COMTRADE dataset, \nand then gives a primer on the tools related to Google matrix analysis such as the trade balance metric and the REGOMAX algorithm.\nIn Section~\\ref{sec:results}, the central results of this papers are presented, which are discussed in \\ref{sec:discussion}. \n\n\\section{Data sets, algorithms and methods}\\label{sec:dataset}\n\nWe use the UN COMTRADE data \\cite{comtrade} for years 2012, 2014, 2016 and 2018 to \nconstruct the trade flows of the multiproduct WTN following the procedure\ndetailed in \\cite{wtn2,wtn3}. This paper gives the results for year 2018 only, others are \nto be found at \\cite{ourwebpage}. Each year is presented by a money matrix, \n$M^{p}_{cc^{\\prime}}$, giving the export flow of product $p$\nfrom country $c^{\\prime}$ to country $c$ (transactions are expressed in USD of current year).\nThe data set is given by $N_{c} = 168$ countries and territories \n(27 EU countries are considered as one country) and $N_{p} = 10$ principal type of products\n(see the lists in \\cite{wtn1,wtn3}). These 10 products are: Food and live animals (0);\nBeverages and tobacco (1); Crude materials, inedible, except fuels (2); Mineral fuels etc (3);\nAnimal and vegetable oils and fats (4); Chemicals and related products, n.e.s. (5);\nBasic manufactures (6); Machinery, transport equipment (7); \nMiscellaneous manufactured articles (8);\nGoods not classified elsewhere (9) (product index $p$ is given in brackets). They belong to \nthe Standard International Trade Classification (SITC Rev. 1)\nThus the total Google matrix $G$ size is given by all system nodes $N=N_c N_p= 1680$\nincluding countries and products.\n\nThe Google matrix $G_{ij}$ of direct trade flows\nis constructed in a standard way described in detail at \\cite{wtn2,wtn3}: monetary trade flows \nfrom a node $j$ to node $i$ are normalized to unity \nfor each column $j$ thus given the matrix $S$ of \nMarkov transitions for trade, the columns of dangling nodes with\nzero transactions are replaced by a column with all elements being $1\/N$. \nThe weight of each product is taken into account via a\ncertain personalized vector taking into account \nthe weight of each product in the global trade volume.\nWe use the damping factor $\\alpha=0.5$. \nThe Google matrix is $G_{ij}=\\alpha S_{ij} + (1-\\alpha) v_i$\nwhere $v_i$ are components of positive column vectors called personalization vectors\nwhich take into account the weight of each product in the global trade \n($\\sum_i v_i=1$).\nWe also construct the matrix $G^*$ for the inverted trade flows.\n\nThe stationary probability distribution \ndescribed by $G$ is given by the PageRank vector\n$P$ with maximal eigenvalue $\\lambda=1$:\n$GP=\\lambda P =P$ \\cite{rmp2015,brin,meyer}. \nIn a similar way, for the inverted flow, described by $G^*$, \nwe have the CheiRank vector $P^*$, \nbeing the eigenvector of $G^* P^* = P^*$. \n PageRank $K$ and CheiRank $K^*$ indexes are obtained from\nmonotonic ordering of probabilities of PageRank vector $P$ and \nof CheiRank vector $P^*$ as\n$P(K)\\ge P(K+1)$ and $P^*(K^*)\\ge P^*(K^*+1)$ with $K,K^*=1,\\ldots,N$.\nThe sums over all products $p$ gives the PageRank and CheiRank \nprobabilities of a given country as $P_c =\\sum_p P_{cp}$ and \n${P^*}_c =\\sum_p {P^*}_{cp}$ (and in a similar way\nproduct probabilities $P_p, {P^*}_p$) \\cite{wtn2,wtn3}.\nThus with these probabilities we obtain the related indexes $K_c, {K^*}_c$.\nWe also define \nfrom import and export trade volume the\n probabilities\n$\\hat{P}_p$, $\\hat{P}^*_p$, $\\hat{P}_c$, $\\hat{P}^*_c$,\n$\\hat{P}_{pc}$, $\\hat{P}^*_{pc}$ and \ncorresponding indexes\n$\\hat{K}_p$, $\\hat{K}^*_p$, $\\hat{K}_c$, $\\hat{K}^*_c$, $\\hat{K}$, $\\hat{K}^*$\n(these import and export probabilities\nare normalized to unity\nby the total import and export volumes, see details in\n\\cite{wtn2,wtn3}). It is useful to note that qualitatively\nPageRank probability is proportional to the volume of ingoing\ntrade flow and CheiRank respectively to outgoing flow.\nThus, we can approximately consider that the\nhigh import gives a high PageRank $P$ probability\nand a high export a high CheiRank $P^*$ probability.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=0.99\\textwidth]{fig1.pdf}\n\t\\caption{Circles with country flags show country positions on\n the plane of PageRank-CheiRank indexes $(K,K^*)$\n (summation is done over all products) (left panel)\n and on the plane of ImportRank-ExportRank $\\hat{K}$, $\\hat{K}^*$ \n from trade volume (right panel);\n data is shown only for index values less than $61$ in year 2018\n \n .} \\label{fig1}\n\\end{figure}\n\n\nAs in \\cite{wtn2,wtn3},\nwe define the trade balance of a given country\nwith PageRank and CheiRank probabilities given by\n$B_c = (P^*_c - P_c)\/(P^*_c + P_c)$.\nAlso we have from ImportRank and ExportRank probabilities\nas $\\hat{B}_c= ({\\hat{P}^*}_c - \\hat{P}_c)\/({\\hat{P}^*}_c + \\hat{P}_c)$.\nThe sensitivity of \ntrade balance $B_c$ to the price of energy or machinery can be obtained \nfrom the change of corresponding money volume\nflow related to SITC Rev.1 code $p=3$ (mineral fuels) or $p=7$ (machinery)\nby multiplying it by $(1+\\delta)$, renormalizing column to unity and \ncomputing all rank probabilities and\nthe derivatives $dB_c\/d\\delta$. \n\nWe also use the REGOMAX algorithm \\cite{politwiki,wtn3}\nto construct the reduced Google matrix $G_R$ \nfor a selected subset of WTN nodes $N_r \\ll N$.\nThis algorithm takes into accounts all transitions of direct and\nindirect pathways happening in the full Google matrix $G$ \nbetween $N_r$ nodes of interest. We use this $G_R$ matrix\nto construct a reduced network of most strong transitions\n(``network of friends'')\nbetween a selection of nodes representing countries and products.\n\nEven if Brexit enter into play in 2021, we use UN COMTRADE data of previous years\nto make a projecting analysis of present and future power of EU composed of 27 countries.\n\nFinally we note that GMA allows to obtain interesting results for \nvarious types of directed networks including Wikipedia \n\\cite{geopolitics,celestin} and protein-protein interaction \n\\cite{zinprotein,signor} networks.\n\n\\begin{table}[]\n \\centering\n \\caption{Top 20 countries of PageRank ($K$), CheiRank ($K^*$), \n ImportRank and ExportRank in 2018.}\n \\begin{tabular}{|c|l|l|l|l|}\n \\hline\n Rank & PageRank & CheiRank & ImportRank & ExportRank \\\\ \\hline\n1 & EU & EU & EU & China \\\\\n2 & USA & China & USA & EU \\\\\n3 & China & USA & China & USA \\\\\n4 & United Kingdom & Japan & Japan & Japan \\\\\n5 & India & Repub Korea & United Kingdom & Repub Korea \\\\\n6 & U Arab Emirates & India & Repub Korea & Russia \\\\\n7 & Japan & Russia & India & United Kingdom \\\\\n8 & Mexico & U Arab Emirates & Canada & Mexico \\\\\n9 & Repub Korea & Singapore & Mexico & Canada \\\\\n10 & Canada & United Kingdom & Singapore & Singapore \\\\\n11 & Singapore & South Africa & Switzerland & Switzerland \\\\\n12 & Switzerland & Thailand & U Arab Emirates & India \\\\\n13 & Turkey & Malaysia & Russia & Malaysia \\\\\n14 & Russia & Canada & Thailand & Australia \\\\\n15 & Australia & Mexico & Viet Nam & U Arab Emirates \\\\\n16 & South Africa & Turkey & Australia & Thailand \\\\\n17 & Thailand & Australia & Turkey & Saudi Arabia \\\\\n18 & Brazil & Switzerland & Malaysia & Viet Nam \\\\\n19 & Saudi Arabia & Brazil & Indonesia & Brazil \\\\\n20 & Malaysia & Saudi Arabia & Brazil & Indonesia \\\\ \\hline\n \\end{tabular}\n\\label{tab1}\n\\end{table}\n\n\\section{Results}\\label{sec:results}\n\\subsection{CheiRank and PageRank of countries}\n\nThe positions of countries on the PageRank-CheiRank $(K,K^*)$ and\nImportRank-ExportRank $(\\hat{K},\\hat{K}^*)$ planes are shown in Fig.~\\ref{fig1}\nand in Table~\\ref{tab1}.\nThese results show a significant difference between these two types of ranking.\nIndeed, EU takes the top PageRank-CheiRank position $K=K^*=1$\nwhile with Export-Import Ranking it has \nonly $\\hat{K}=1; \\hat{K}^*=2$, with USA at $\\hat{K}=2, \\hat{K}^*=3$\nand China at $\\hat{K}=3, \\hat{K}^*=1$. Thus EU takes the leading positions\nin the GMA frame which takes into account the muliplicity of trade transactions\nand characterizes the robust features of EU trade relations.\nAlso GMA shows that UK position is significantly weakened compared to IEA description\n(thus UK moves from $K^*=7$ in IEA to $K^*=10$ in GMA). From this data, we see also examples\nof other countries that significantly improve there rank positions in GMA\nframe compared to IEA: India ($K=5$, $K^*=6$, $\\hat{K}=7$, $\\hat{K}^*=12$), \nUnited Arab Emirates ($K=6$, $K^*=8$, $\\hat{K}=12$, $\\hat{K}^*=15$), \nSouth Africa ($K=16$, $K^*=11$, $\\hat{K}=23$, $\\hat{K}^*=23$). \nWe attribute this to well developed, deep \nand broad trade network of these countries which are well captured by GMA in contrast\nto IEA. Indeed, IEA only measures the volume of direct trade exchanges, while GMA characterises \nthe multiplicity of trade chains in the world.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{fig2.pdf}\n\t\\caption{World map of trade balance of countries \n$B_c = ({P_c}^* - P_c)\/({P_c}^* + P_c)$.\nTop: trade balance values are computed from the trade volume of Export-Import;\nbottom: trade balance values are computed from PageRank\nand CheiRank vectors; $B_c$ values are marked by color with the corresponding \ncolor bar marked by $j$; \ncountries absent in the UN COMTRADE report are marked by black color \n(here and in other Figs).}\\label{fig2}\n\\end{figure}\n\n\\subsection{Trade balance and its sensitivity to product prices}\n\nThe trade balance of countries in IEA and GMA frames is shown in Fig.~\\ref{fig2}.\nThe countries with 3 strongest positive balance are: \nEquatorial Guinea ($B_c=0.732$), Congo ($B_c=0.645$), Turkmenistan ($B_c=0.623$) in IEA and\nChina ($B_c=0.307$), Japan ($B_c= 0.244$), Russia ($B_c= 0.188$) in GMA.\nWe see that IEA marks top countries which have no significant world power\nwhile GMA marks countries with real significant world influence.\nFor EU and UK we have respectively $B_c=-0.015; 0.020$ (EU) and \n$B_c=-0.178; -0.187$ (UK) in IEA; GMA.\nThus the UK trade balance is significantly reduced in GMA\ncorresponding to a loss of network trade influence of UK\nin agreement with data of Fig.~\\ref{fig1} and Table~\\ref{tab1}.\n(We note that the balance variation bounds in GMA are smaller compared to IEA;\nwe attribute this to the fact of multiplicity of transactions \nin GMA that smooth various fluctuations which are more typical for IEA).\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig3.pdf}\n\t\\caption{Sensitivity of country balance \n$dB_c\/d\\delta_s$ for product $s=3$ (mineral fuels).\nTop: probabilities are from the trade volume of Export-Import;\nbottom: probabilities are from PageRank\nand CheiRank vectors. Color bar marked by $j$ \ngives sensitivity.} \\label{fig3}\n\\end{figure}\n\nThe balance sensitivity $d B_c\/d \\delta_s$ to product $s=3$\n(mineral fuels (with strong petroleum and gas contribution))\nis shown in Fig.~\\ref{fig3}. The top 3 strongest \npositive sensitivities $d B_c\/d \\delta_s$ \nare found for Algeria (0.431), Brunei (0.415), South Sudan (0.411) in IEA\nand Saudi Arabia (0.174), Russia (0.161), Kazakhstan (0.126) in GMA.\nThe results of GMA are rather natural since Saudi Arabia, Russia and Kazakhstan are central petroleum producers. \nIt is worth noting that GMA ranks Iraq at the 4th position. \nThe 3 strongest negative sensitivities are Zimbabwe (-0.137), Nauru (-0.131), \nJapan (-0.106), in IEA\nand Japan (-0.066), Korea (-0.062), Zimbabwe (-0.058), in GMA.\nFor China, India we have $d B_c\/d \\delta_s$ values being respectively:\n-0.073, -0.086 in IEA and -0.056, 0.010 in GMA. This shows that the trade network\nof India is more stable to price variations of product $s=3$.\nThese results demonstrate that GMA selects more globally influential countries.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig4.pdf}\n\t\\caption{Same as in Fig.~\\ref{fig3} but\nfor product $s=7$ (machinery).} \\label{fig4}\n\\end{figure}\n\nThe balance sensitivity $d B_c\/d \\delta_s$ to product $s=7$\n(machinery) is shown in Fig.~\\ref{fig4}. \nHere the top 3 strongest \npositive sensitivities $d B_c\/d \\delta_s$ are found\nin both IEA and GMA for Japan (respectively 0.167, 0.151), \nRepub. Korea (0.143, 0.097), Philippines (0.130, 0.091).\nThe 3 strongest negative sensitivities are Brunei (-0.210), Iran (-0.202), \nUzbekistan (-0.190) in IEA\nand Russia (-0.138), Kazakhstan (-0.102), Argentina (-0.097) in GMA.\nThus we again see that GMA selects more globally influential countries. \nThe sensitivity $d B_c\/d \\delta_s$ values for EU, UK, China, Russia, USA are:\n EU (0.048), UK (0.006), China (0.065), Russia (-0.170), USA (-0.027) in IEA;\n EU (0.000), UK (0.024), China (0.077), Russia (-0.138), USA (-0.018) in GMA.\nLatter GMA results show that even if machinery product ($s=7$) is very important for EU \nthe network power of trade with this product becomes dominated by Asian countries\nJapan, Repub. Korea, China, Philippines; \nin this aspect the position of UK is slightly better than EU. \n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig5.pdf}\n\t\\caption{Sensitivity of country balance \n$dB_c\/d\\delta_{cs}$ for product price $s=7$ (machinery) \nof EU (top left), USA (top right), China (bottom left)\nand $s=3$ (mineral fuel) of Russia (bottom right);\n$B_c$ is computed from PageRank and CheiRank vectors;\nsensitivity values are marked by color with the corresponding \ncolor bar marked by $j$. \nFor EU, USA, China, Russia we have $dB_c\/d\\delta_{cs} = 0.11, 0.11, 0.14, 0.12$\nrespectively, these values are \nmarked by separate magenta color to highlight sensitivity \nof other countries in a better way.} \\label{fig5}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig6.pdf}\n\t\\caption{Network trade structure between \nEU, USA (US), China (CN), Russia (RU) \nwith 10 products. \nNetwork is obtained from the reduced\nGoogle matrix $G_R$ (left) and ${G^*}_R$ (right)\nby tracing four strongest outgoing links\n(similar to 4 ``best friends'').\nCountries are shown by circles with two letters of country\nand product index listed in Section 2. The arrow direction \nfrom node $A$ to node $B$ means that $B$ imports from $A$ (left)\nand $B$ exports to $A$ (right).\nAll $40$ nodes are shown.\n} \\label{fig6}\n\\end{figure}\n\nIn Figs.~\\ref{fig3} and \\ref{fig4}, we have considered the sensitivity of \ncountry balance to a global price of a specific product (mineral fuel $s=3$ or machinery $s=7$). \nIn contrast, with GMA, we can also obtain the sensitivity of country balance to the price of\nproducts originating from a specific country. Such results are shown in Fig.~\\ref{fig5}.\nThey show that machinery ($s=7$) of EU gives a significant positive balance sensitivity \nfor UK and negative for Russia. This indicates a strong dependence of Russia \nfrom EU machinery. Machinery of USA gives strong positive effect for Mexico and Canada\nwith a negative effect for EU, Russia, Brazil, Argentina.\nMachinery of China gives positive sensitivity for Asian countries\n(Repub. Korea, Japan, Philippines) and significant negative effect for Mexico.\nMineral fuels ($s=3$) of Russia gives positive effect for Kazakhstan, Uzbekistan, Ukraine\n(former USSR republics) and negative effect for competing petroleum and gas producers\nNorway and Algeria.\n\n\\subsection{Network structure of trade from reduced Google matrix}\n\nThe network structure for 40 nodes of 10 products of EU, USA, China and Russia is \nshown in Fig.~\\ref{fig6}. It is obtained from the reduced Google matrix of $N_r=40$ nodes\nof global WTN network with $N=1680$ nodes on the basis of REGOMAX algorithm\nwhich takes into account all pathways between $N_r$ nodes via the global network of $N$ nodes.\nThe networks are shown for the direct ($G$ matrix) and\ninverted ($G^*$ matrix) trade flows. For each node we show only 4 strongest outgoing \nlinks (trade matrix elements) that heuristically can be considered as the four ``best friends''. \nThe resulting network structure clearly shows\nthe central dominant role of machinery product. For ingoing flows (import direction) of $G_R$ \nthe central dominance of machinery for USA and EU is directly visible while \nfor outgoing flows (export direction), machinery of EU and China dominate exports.\n\nIt is interesting to note that the network influence of EU with 27 countries\nis somewhat similar to the one constituted by a kernel of 9 dominant EU countries (KEU9)\n(being Austria, Belgium, France, Germany, Italy,\nLuxembourg, Netherlands, Portugal, Spain) discussed in \\cite{keu9}. This shows\nthe leading role played by these KEU9 countries in the world trade influence of EU.\n\nFinally we note that additional data with figures and tables is available at \\cite{ourwebpage}.\n\n\\section{Discussion}\\label{sec:discussion}\n\nWe presented the Google matrix analysis of multiproduct WTN obtained from UN COMTRADE database\nin recent years. In contrast to the legacy Import-Export characterization of trade,\nthis new approach captures multiplicity of trade transactions between world countries\nand highlights in a better way the \nglobal significance and influence of trade relations between specific countries and products.\nThe Google matrix analysis clearly shows that the dominant position\nin WTN is taken by the EU of 27 countries despite the leave of UK after Brexit.\nThis result demonstrates the robust structure of worldwide EU trade.\nIt is in contrast with the usual Import-Export analysis in which USA and China\nare considered as main players. We also see that machinery and mineral fuels\nproducts play a dominant role in the international trade.\nThe Google matrix analysis stresses the growing dominance of \nmachinery products of Asian countries (China, Japan, Republic of Korea).\n\nWe hope that the further development of Google matrix analysis \nof world trade will bring new insights in this complex system of world economy.\n\n{\\it Acknowledgments:}\nWe thank Leonardo Ermann for useful discussions.\nThis research has been partially supported through the grant\nNANOX $N^\\circ$ ANR-17-EURE-0009 (project MTDINA) in the frame \nof the {\\it Programme des Investissements d'Avenir, France} and\nin part by APR 2019 call of University of Toulouse and by \nRegion Occitanie (project GoIA).\nWe thank UN COMTRADE for providing us a friendly access\nto their detailed database.\n\n\n\n\n\n\\section{#1}\n \\addcontentsline{lof}{section}{%\n Section \\thesection: #1 \\vspace{10pt}\n }\n}\n\n\n\n\n\\renewcommand{\\figurename}{SupFig.}\n\n\n\\date{February 2021}\n\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\n\\begin{document}\n\n\\title{Supplementary Material for: \\\\ Post-Brexit power of European Union \\\\ from the world trade network analysis}\n\n\\author{Justin Loye\\inst{1,2} \\and\nKatia Jaffr\\`es-Runser\\inst{3}\n\\and\nDima L. Shepelyansky\\inst{2}\n}\n\\authorrunning{J.Loye et al.}\n\\titlerunning{Supplementary Material for: Post-Brexit power of European Union from the WTN}\n\n\\institute{Institut de Recherche en Informatique de Toulouse, \\\\\nUniversit\\'e de Toulouse, UPS, 31062 Toulouse, France\\\\\n\\email{justin.loye@irit.fr}\\\\\n\\and\nLaboratoire de Physique Th\\'eorique, \nUniversit\\'e de Toulouse, CNRS, UPS, 31062 Toulouse, France\\\\\n\\email{dima@irsamc.ups-tlse.fr}\\\\\n\\and\nInstitut de Recherche en Informatique de Toulouse, \\\\\nUniversit\\'e de Toulouse, Toulouse INP, 31071 Toulouse, France\\\\\n\\email{katia.jaffresrunser@irit.fr}\\\\\n}\n\\maketitle \n\n\\vspace{1.0cm}\n\nWe present additional results SupFig.~\\ref{fig:KKstar_2012_2016}, \\ref{fig:balance2012_2016}, \\ref{fig:importations},\n\\ref{fig:exportations}, \\ref{fig:dbalance_petroleum_2012_2016} and \\ref{fig:dbalance_machinery_2012_2016}.\n\nSupFig.~\\ref{fig:KKstar_2012_2016}, \\ref{fig:balance2012_2016}, \\ref{fig:dbalance_petroleum_2012_2016} and \\ref{fig:dbalance_machinery_2012_2016}\nreproduce for years 2012, 2014, 2016 respectively Fig.1, Fig.2, Fig.3 and Fig.4 of the main article. \nIn SupFig.~\\ref{fig:importations} and \\ref{fig:exportations}, we respectively show the volume of importation and exportation of countries in the world trade network.\n\n\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.85\\textwidth]{figS1.pdf}\n \\caption{Circles with country flags show country positions on\n the plane of PageRank-CheiRank indexes $(K,K^*)$\n (summation is done over all products) (left panel)\n and on the plane of ImportRank-ExportRank $\\hat{K}$, $\\hat{K}^*$ \n from trade volume (right panel);\n data is shown only for index values less than $61$.\n \n First, second and third row correspond to years 2012, 2014, 2016.}\n \\label{fig:KKstar_2012_2016}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.96\\textwidth]{figS2.pdf}\n \\caption{World map of trade balance of countries \n $B_c = ({P_c}^* - P_c)\/({P_c}^* + P_c)$.\n Left: probabilities are from PageRank and CheiRank vectors;\n Right: probabilities are from the trade volume of Export-Import;\n $B_c$ values are marked by color with the corresponding \n color bar marked by $j$; \n \n \n countries absent in the UN COMTRADE report are marked by black color \n (here and in other SupFigs).\n First, second and third row correspond to year 2012, 2014, 2016.}\n \\label{fig:balance2012_2016}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{figS3.pdf}\n \\caption{World map of countries importation volume in dollar.\n Values lower than $10^8$ are clipped for better visibility.\n First, second, third and fourth row correspond to years 2012, 2014, 2016, 2018.\n }\n \\label{fig:importations}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{figS4.pdf}\n \\caption{World map of countries exportation volume in dollar.\n Values lower than $10^8$ are clipped for better visibility.\n First, second, third and fourth row correspond to years 2012, 2014, 2016, 2018.\n }\n \\label{fig:exportations}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.96\\textwidth]{figS5.pdf}\n \\caption{Sensitivity of country balance \n $dB_c\/d\\delta_s$ for product $s=3$ (mineral fuels).\n Left: probabilities are from PageRank and CheiRank vectors;\n Right: probabilities are from the trade volume of Export-Import.\n Color bar marked by $j$ \n \n gives sensitivity..\n First, second and third row correspond to year 2012, 2014, 2016.}\n \\label{fig:dbalance_petroleum_2012_2016}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.96\\textwidth]{figS6.pdf}\n \\caption{Same as in Fig.~\\ref{fig:dbalance_petroleum_2012_2016} but\n for product $s=7$ (machinery).}\n \\label{fig:dbalance_machinery_2012_2016}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe spectral heat content represents the total heat \nin a domain $D$ with Dirichlet boundary condition when the initial temperature is 1.\nThe spectral heat content for Brownian motions has been studied extensively.\nThe spectral heat content for isotropic stable processes was first studied in \\cite{Val2017}. \nSince then, considerable progress has been made toward understanding the asymptotic behavior of the spectral heat content for other L\\'evy processes (see \\cite{Val2016, GPS19, P20, PS19}).\n\n\nThe following conjecture about the spectral heat content for isotropic $\\alpha$-stable\nprocesses, $\\alpha\\in (0,2)$, over bounded $C^{1,1}$ open sets (see Section \\ref{section:preliminaries} for the definition of $C^{1,1}$ open sets) was made in \\cite{Val2017}: \nAs $t\\downarrow 0$,\n\\begin{equation}\\label{eqn:Val conj}\nQ_{D}^{(\\alpha)}(t)=\n\\begin{cases}\n|D|-c_{1}|\\partial D|t^{1\/\\alpha} +O(t), \\quad \\alpha\\in (1,2),\\\\\n|D|-c_{2}|\\partial D|t\\ln(1\/t) +O(t), \\quad \\alpha=1,\\\\\n|D|-c_{3}\\text{Per}_{\\alpha}(D)t +o(t), \\quad \\alpha\\in (0,1),\n\\end{cases}\n\\end{equation}\nwhere $c_i, i=1, 2, 3,$ are constants and \n\\begin{equation}\\label{eqn:fractional perimeter}\n\\text{Per}_{\\alpha}(D):=\\int_{D}\\int_{D^{c}}\\frac{A_{d,\\alpha}}{|x-y|^{d+\\alpha}}dydx,\n\\quad \\mbox{ with }A_{d,\\alpha}=\\frac{\\alpha\\Gamma(\\frac{d+\\alpha}{2})}{2^{1-\\alpha}\\pi^{d\/2}\\Gamma(1-\\frac{\\alpha}{2})},\n\\end{equation}\nis the $\\alpha$-fractional perimeter of $D$.\nThis conjecture was resolved in dimension 1 in \\cite{Val2016} (actually, a slightly weaker version, with the error term being $o(t^{1\/\\alpha})$ in the case $\\alpha\\in (1, 2)$ and $o(t\\ln(1\/t))$ in the case $\\alpha=1$, of the conjecture was proved there).\nWe note that in \\cite{Val2017} the author \nconjectured, and also provided strong evidence, that the spectral heat content for isotropic $\\alpha$-stable processes with $\\alpha\\in (0,2)$ must have an asymptotic expansion of the form as \\eqref{eqn:Val conj} for all dimensions $d\\geq2$, \nbut exact expressions for the coefficients $c_{i}$ were not provided.\nThen a two-term asymptotic expansion of the spectral heat content for L\\'evy processes of bounded variation in ${\\mathbb R}^{d}$ was established in \\cite{GPS19}.\nSince $\\alpha$-stable processes are of bounded variation if and only if $\\alpha\\in (0,1)$, \nthe result in \\cite{GPS19} proves \\eqref{eqn:Val conj} for $\\alpha\\in (0,1)$. \nThe purpose of this paper is to resolve the conjecture above for $\\alpha \\in [1,2)$ and $d\\geq 2$. \nIn fact, our result is slightly weaker than \\eqref{eqn:Val conj} since the error term is $o(t^{1\/\\alpha})$ for $\\alpha\\in (1, 2)$ and $o(t\\ln(1\/t))$ for $\\alpha=1$.\nWe also find explicit expressions for the constants $c_{1}$ and $c_{2}$.\nThe main results of this paper are Theorem \\ref{thm:main12} for $\\alpha\\in (1,2)$ and Theorem \\ref{thm:main1} for $\\alpha=1$. \nCombining Theorems \\ref{thm:main12} and \\ref{thm:main1} with \\cite[Corollary 3.5]{GPS19}, the asymptotic behavior of the spectral heat content for isotropic $\\alpha$-stable processes in bounded $C^{1,1}$ open sets $D$ can be stated as follows: \n\n\\begin{thm}\\label{thm:main}\nLet $D$ be a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$, $d\\geq 2$ and \nlet\n$$\nf_{\\alpha}(t)=\n\\begin{cases}\nt^{1\/\\alpha} &\\text{ if } \\alpha\\in (1,2),\\\\\nt\\ln(1\/t) &\\text{ if } \\alpha=1,\\\\\nt &\\text{ if } \\alpha\\in (0,1).\n\\end{cases}\n$$\nThen, we have\n\\begin{equation}\\label{eqn:main result}\n\\lim_{t\\to 0}\\frac{|D|-Q_{D}^{(\\alpha)}(t)}{f_{\\alpha}(t)}=\n\\begin{cases}\n{\\mathbb E}[\\overline{Y}_{1}^{(\\alpha)}]|\\partial D| &\\text{if } \\alpha\\in (1,2),\\\\\n\\frac{|\\partial D|}{\\pi} &\\text{if }\\alpha=1,\\\\\n\\rm{Per}_{\\alpha}(D), &\\text{if } \n\\alpha\\in (0,1),\n\\end{cases}\n\\end{equation}\nwhere $\\overline{Y}_{t}^{(\\alpha)}=\\sup_{s\\le t}Y^{(\\alpha)}_s$ stands for the running supremum of a 1 dimensional symmetric $\\alpha$-stable process $Y^{(\\alpha)}_t$ and \n$\\rm{Per}_{\\alpha}(D)$ as defined in \\eqref{eqn:fractional perimeter}.\n\\end{thm}\nWe note that \\eqref{eqn:main result} is exactly the same form as \\cite[Theorem 1.1]{Val2016} if one interprets $|\\partial D|=2$ when $D$ is a bounded open interval in ${\\mathbb R}$, but the proof for $d\\geq 2$ is very different from the one dimensional case and much more challenging.\n\nThe two-term asymptotic expansion of the spectral heat content for Brownian motion was proved in \\cite{BD89}. \nThe crucial ingredient in \\cite{BD89} is the fact that \nindividual components of Brownian motion are independent.\nFor isotropic $\\alpha$-stable processes, $\\alpha\\in (0, 2)$, individual components are not independent and the technique in \\cite{BD89} no longer works.\n When $\\alpha\\in (1,2)$,\nwe establish the lower bound for the heat loss $|D|-Q_{D}^{(\\alpha)}(t)$ by considering the most efficient way of exiting $D$ (see Lemma \\ref{lemma:lb}).\nIn order to establish the upper bound, we approximate the heat loss \n$|D|-Q_{D}^{(\\alpha)}(t)$ by the heat loss of the half-space \n$|D|-Q_{H_{x}}^{(\\alpha)}(t)$ for \n$x$ near $\\partial D$ (see Proposition \\ref{prop:ub12}) \nand show that the approximation error is of order $o(t^{1\/\\alpha})$ in Lemma \\ref{lemma:near boundary}, \nwhich is similar to tools exposed in the trace estimate results in \\cite{BK08, PS14}.\nHowever, these tools do not work when $\\alpha=1$ due to the non-integrability of \n${\\mathbb P}(\\overline{Y}^{(1)}_{1}>u)$ over $(1,\\infty)$, where $ \\overline{Y}^{(1)}_t$ stands for the\nsupremum of the Cauchy process up to time $t$, \nand the proof for $\\alpha=1$ requires new ideas and is considerably more difficult. \n\nIn case of $\\alpha=1$, we prove that the coefficient of the second term \nof the asymptotic expansion of $Q_{D}^{(1)}(t)$ is $-\\frac{|\\partial D|}{\\pi}$, \nwhich is the same as for the regular heat content $H^{(1)}_{D}(t)$ \nwhich represents the total heat in $D$ without the Dirichlet exterior condition \n(see \\eqref{e:rhc} below for the definition of regular heat content and \\cite[Theorem 1.2]{Val2017} \nfor the two-term asymptotic expansion for the regular heat content $H_{D}^{(1)}(t)$). \nIf there is no Dirichlet exterior condition on $D^{c}$, or equivalently if the heat moves freely in and out of $D$,\nthe heat loss of the regular heat content must be smaller than that of the spectral heat content and we obtain the lower bound for free in \\eqref{eqn:Cauchy lb}. The proof for the upper bound is much more demanding. \nThe crucial ingredient for the upper bound is the spectral heat content for subordinate killed Brownian motions in \\cite{PS19}. \nAn isotropic $\\alpha$-stable process $X_{t}^{(\\alpha)}$ can be realized as a subordinate Brownian motion $W_{S_{t}^{(\\alpha\/2)}}$, where $S_{t}^{(\\alpha\/2)}$ is an independent $(\\alpha\/2)$-stable subordinator.\nHence, the spectral heat content $Q_{D}^{(\\alpha)}(t)$ is the spectral heat content for the \\textit{killed subordinate Brownian motion} via the independent $(\\alpha\/2)$-stable subordinator $S_{t}^{(\\alpha\/2)}$. \nWhen one reverses the order of killing and subordination, one obtains the \\textit{subordinate killed Brownian motion}. \nThis is the process obtained by subordinating the killed Brownian motion in $D$ via the independent $(\\alpha\/2)$-stable subordinator $S_{t}^{(\\alpha\/2)}$. \nLet $\\widetilde{Q}_{D}^{(\\alpha)}(t)$ be the spectral heat content of the subordinate killed Brownian motion in $D$\n(see \\eqref{eqn:SKBM} for the precise definition).\nBy construction,\n $\\widetilde{Q}_{D}^{(\\alpha)}(t)$ is always smaller than $Q_{D}^{(\\alpha)}(t)$, \nor equivalently the heat loss $|D|-\\widetilde{Q}_{D}^{(\\alpha)}(t)$ of the subordinate killed Brownian motion provides a natural upper bound for \nthat of the killed subordinate Brownian motion $|D|-Q_{D}^{(\\alpha)}(t)$. \nWe use two independent $\\alpha\/2$ and $\\beta\/2$ stable subordinators \nwith $\\alpha\\beta=2$ \nand consider the $\\alpha$-stable process $X^{(\\alpha)}_{t}=W_{S_{t}^{(\\alpha\/2)}}$ and killed it upon exiting $D$. \nThen we time-change the killed $\\alpha$-stable process by the $(\\beta\/2)$-stable subordinator. \nBy using the heat loss of the resulting process, we obtain in Lemma \n\\ref{lemma:Cauchy ub} that, for $\\alpha\\in (1,2)$, \n$$\n\\limsup_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}\\leq\n\\frac{|\\partial D|}{\\pi} +\n\\frac{\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}u, Y_{1}^{(\\alpha)}u)-{\\mathbb P}(Y^{(\\alpha)}_{1}\\geq u)\n$$\nand it can be shown that \n${\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)\\sim {\\mathbb P}(Y^{(\\alpha)}_{1}\\geq u) \\sim cu^{-\\alpha}$ \nas $u\\to \\infty$, \nhence\n$\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)du$ and $\\int_{0}^{\\infty}{\\mathbb P}(Y^{(\\alpha)}_{1}\\geq u)du$ \nshould be of order $\\frac{1}{\\alpha-1}$ as $\\alpha\\downarrow 1$.\nWe show that\n$\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)du$ and $\\int_{0}^{\\infty}{\\mathbb P}(Y^{(\\alpha)}_{1}\\geq u)du$ \nhave \\textit{the exactly same leading coefficients} and this gives a cancellation of the main terms of order $\\frac{1}{\\alpha-1}$. \nWe still need to show that the sub-leading terms are of order $o(\\alpha-1)$ as $\\alpha\\downarrow 1$ and we show this by establishing uniform heat kernel upper and lower bounds in Lemma \\ref{lemma:uniform ub} \nfor the heat kernel of the supremum process $\\overline{Y}_{t}^{(\\alpha)}$ \nand Lemma \\ref{lemma:uniform lb} for the heat kernel of $Y_{t}^{(\\alpha)}$.\n\n\nThis paper deals with asymptotic behavior of the spectral heat content for isotropic $\\alpha$-stable processes. \nIt is natural and interesting to try to find the asymptotic behavior of the spectral heat content for more general L\\'evy processes. We intend to deal with this topic in a future project.\nIn the recent paper \\cite{P20}, a three-term asymptotic expansion of the spectral heat content of 1 dimensional symmetric $\\alpha$-stable processes, $\\alpha\\in [1,2)$, was established.\nWe believe that a similar result should hold true for $d\\geq 2$.\n\nThe organization of this paper is as follows. \nIn Section \\ref{section:preliminaries}, we introduce the setup. In Section \\ref{section:alpha12},\nwe deal with the case $\\alpha\\in (1,2)$ and the main result of that section is Theorem \\ref{thm:main12}. \nThe case $\\alpha=1$ is dealt with in Section \\ref{section:alpha1} and the main result there is Theorem \\ref{thm:main1}. \nIn this paper, we use $c_i$ to denote constants whose values are unimportant and may change from one appearance to another.\nThe notation ${\\mathbb P}_{x}$ stands for the law of the underlying processes started at $x\\in {\\mathbb R}$, and ${\\mathbb E}_{x}$ stands for expectation with respect to ${\\mathbb P}_{x}$. For simplicity, we use ${\\mathbb P}={\\mathbb P}_{0}$ and ${\\mathbb E}={\\mathbb E}_{0}$.\n\n\n\\section{Preliminaries}\\label{section:preliminaries}\n\nIn this paper, unless explicitly stated otherwise, we assume $d\\geq 2$.\nLet $X_{t}^{(\\alpha)}$, $\\alpha\\in (0, 2]$, be an isotropic $\\alpha$-stable process with \n$$\n{\\mathbb E}[e^{i\\xi X^{(\\alpha)}_{t}}]=e^{-t|\\xi|^{\\alpha}}, \\quad \\xi \\in {\\mathbb R}^{d}, \\alpha\\in (0,2].\n$$\n$X_{t}^{(2)}$ is a Brownian motion $W_t$ with transition density given by $(4\\pi t)^{-d\/2}e^{-\\frac{|x|^2}{4t}}$. \nLet $S_{t}^{(\\alpha\/2)}$, $\\alpha\\in (0,2)$, be an $(\\alpha\/2)$-stable subordinator with\n$$\n{\\mathbb E}[e^{-\\lambda S_{t}^{(\\alpha\/2)}}]=e^{-t \\lambda^{\\alpha\/2}}, \\quad \\lambda>0.\n$$\nAssume that $S_{t}^{(\\alpha\/2)}$ is independent of the Brownian motion $W_t$. \nThen, the subordinate Brownian motion $W_{S_{t}^{(\\alpha\/2)}}$ is a realization of the process $X_{t}^{(\\alpha)}$.\nWe will reserve $Y_{t}^{(\\alpha)}$ for the 1 dimensional symmetric $\\alpha$-stable process.\nWe define the running supremum process $\\overline{Y}^{(\\alpha)}_{t}$ of $Y_{t}^{(\\alpha)}$ by\n\\begin{equation}\\label{eqn:running}\n\\overline{Y}^{(\\alpha)}_{t}=\\sup\\{Y^{(\\alpha)}_{u} : 0\\leq u\\leq t\\}.\n\\end{equation}\n\nRecall that an open set $D$ in ${\\mathbb R}^d$ is said to be a $C^{1, 1}$ open set if \nthere exist a localization radius $R_0>0$ and a constant $\\Lambda_0>0$ such that, \nfor every $z\\in \\partial D$, there exist a $C^{1, 1}$ function $\\phi=\\phi_z: {\\mathbb R}^{d-1}\\to {\\mathbb R}$ satisfying $\\phi(0)=0$, $\\nabla \\phi(0)=(0, \\cdots, 0)$, $\\|\\nabla\\phi\\|_\\infty\\le \\Lambda_0$,\n$|\\nabla \\phi(x)-\\nabla \\phi(y)|\\le \\Lambda_0 |x-y|$ and an orthonormal coordinate system $CS_z: y=(\\widetilde y, y_d)$\nwith origin at $z$ such that\n$$\nB(z, R_0)\\cap D=B(z, R_0)\\cap \\{ y=(\\widetilde y, y_d) \\mbox{ in } CS_z: y_d>\\phi(\\widetilde y)\\}.\n$$\nThe pair $(R_0, \\Lambda_0)$ is called the $C^{1,1}$ characteristics of the $C^{1, 1}$ open set $D$. \nIt is well known that any $C^{1, 1}$ open set $D$ in ${\\mathbb R}^d$ satisfies the uniform interior and exterior $R$-ball condition: for any $z\\in\\partial D$, there exist balls $B_1$ and $B_2$ of radii $R$ with $B_1\\subset D$, $B_2\\subset{\\mathbb R}^d\\setminus\\overline{D}$ and $\\partial B_1\\cap\\partial B_2=\\{z\\}$.\n\nWe recall from \\cite{BD89} a useful fact about open sets $D$ satisfying the uniform interior and exterior $R$-ball condition.\nLet $D_{q}=\\{x\\in D : \\text{dist}(x, \\partial D) >q\\}$.\nWe will use $\\partial D_{q}$ denote the portion of the boundary of $D_q$ contained in $D$, that is, $\\partial D_{q}=\\{x\\in D: \\text{dist}(x, \\partial D)=q\\}$.\nIt follows from \\cite[Lemma 6.7]{BD89}\nthat\n\\begin{equation}\\label{eqn:stability}\n|\\partial D|\\left(\\frac{R-q}{R}\\right)^{d-1}\\leq |\\partial D_{q}|\\leq |\\partial D|\\left(\\frac{R}{R-q}\\right)^{d-1}, \\quad 0< q0: X^{(\\alpha)}_{t}\\notin D\\}$ be the first exit time of $X^{(\\alpha)}_{t}$ from $D$. \nThe spectral heat content of $D$ for $X^{(\\alpha)}_{t}$ is defined to be\n$$\nQ_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}>t)dx.\n$$\nThe (regular) heat content $H_{D}^{(\\alpha)}(t)$ of $D$ for $X^{(\\alpha)}_{t}$ is defined to be\n\\begin{equation}\\label{e:rhc}\nH_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(X_{t}^{(\\alpha)}\\in D)dx.\n\\end{equation}\nThe spectral heat content $\\widetilde{Q}_{D}^{(\\alpha)}(t)$ of $D$ for the subordinate killed Brownian motion is defined as \n\\begin{equation}\\label{eqn:SKBM}\n\\widetilde{Q}_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(2)}>S_{t}^{(\\alpha\/2)})dx.\n\\end{equation}\nNote that we always have \n\\begin{equation}\\label{eqn:two heat contents}\n\\{\\tau_{D}^{(2)}>S_{t}^{(\\alpha\/2)}\\} \\subset \\{\\tau_{D}^{(\\alpha)}>t\\}\\subset \\{X_{t}^{(\\alpha)}\\in D\\}.\n\\end{equation}\nWe sometimes use the terminology \\textit{heat loss} and this will mean either\n$$\n|D|-Q_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}\\leq t)dx,\n$$\nor \n$$\n|D|-\\widetilde{Q}_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(2)}\\leq S_{t}^{(\\alpha\/2)})dx,\n$$\ndepending on which process we are dealing with. \nIntuitively, these quantities represent the total heat loss caused by heat particles jumping out of $D$ up to time $t$.\nFrom \\eqref{eqn:two heat contents}, we have \n$$\n|D|- H_{D}^{(\\alpha)}(t)\\leq |D|-Q_{D}^{(\\alpha)}(t) \\leq |D|-\\widetilde{Q}_{D}^{(\\alpha)}(t), \\quad t>0.\n$$\n\n\n\n\\section{The case $\\alpha\\in (1,2)$}\\label{section:alpha12}\nThroughout this section, we assume $\\alpha\\in(1,2)$.\nFor any $x\\in D$, we use $\\delta_D(x)$ to denote the distance between $x$ and $\\partial D$.\nWe start with a lower bound. \nRecall that $D_{q}=\\{x\\in D : \\text{dist}(x, \\partial D) >q\\}$.\n\n\n\n\\begin{lemma}\\label{lemma:lb}\nLet $D$ be a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$. Then, we have\n$$\n\\liminf_{t\\to 0}\\frac{|D|-Q^{(\\alpha)}_{D}(t)}{t^{1\/\\alpha}}\\geq |\\partial D|\n{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}],\n$$\nwhere $\\overline{Y}_{t}^{(\\alpha)}$ is defined in \\eqref{eqn:running}.\n\\end{lemma}\n\\noindent{\\bf Proof.} \nLet $D$ satisfy the uniform interior and exterior $R$-ball condition.\nFix $a\\leq R\/2$. \nFor $x\\in D\\setminus D_{a}$, let \n$z_x\\in \\partial D$ be such that $|x-z_x|=\\delta_D(x)$ and let ${\\bf n}_{z_x}$ be the outward unit normal vector to the boundary $\\partial D$ at the point $z_x$.\n\nFor $X$ starting from $x$, we define \n$Y^{(\\alpha)}_{t}:=(X_{t}^{(\\alpha)}-x)\\cdot {\\bf n}_{z_x}$, where $\\cdot$ stands for usual scalar product in ${\\mathbb R}^{d}$. Obviously, $Y^{(\\alpha)}_{0}=0$.\nNote that the characteristic exponent of $Y^{(\\alpha)}_{t}$ is given by \n$$\n{\\mathbb E}[e^{i\\eta Y^{(\\alpha)}_{t}}]=\n{\\mathbb E}_x[e^{i\\eta (X_{t}^{(\\alpha)}-x)\\cdot{\\bf n}_{z_x}}]\n=e^{-t|\\eta|^{\\alpha}}, \n\\quad \\eta \\in {\\mathbb R},\n$$\nand this shows that $Y^{(\\alpha)}_{t}$ is a one dimensional stable process starting from 0.\nLet $r\\le a$ and $x\\in \\partial D_{r}$, and let $H_{x}$ be the half-space containing the interior $R$-ball at the point $z_x$ and tangent to $\\partial D$ at $z_x$.\nWhen the process $X$ starts from $x$, we have\n$$\n\\{\\overline{Y}_{t}^{(\\alpha)} >r\\}=\\{\\tau_{H_x}^{(\\alpha)}\\leq t\\}\\subset \\{\\tau_{D}^{(\\alpha)} \\leq t\\} \n$$\nand this implies \n$$\n{\\mathbb P}(\\overline{Y}_{t}^{(\\alpha)} >r)\\leq {\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)} \\leq t).\n$$\nHence, by the coarea formula\nand \\eqref{eqn:stability}, we have\n\\begin{align*}\n&|D|-Q^{(\\alpha)}_{D}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dx\n\\geq \\int_{D\\setminus D_{a}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dx\\\\\n&=\\int_{0}^{a}\\int_{\\partial D_{r}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dSdr\n\\geq \\int_{0}^{a}\\int_{\\partial D_{r}}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{t}>r)dSdr\\\\\n&=\\int_{0}^{a}|\\partial D_{r}|{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{t}>r)dr\n\\geq |\\partial D|\\left(\\frac{R-a}{R}\\right)^{d-1}\\int_{0}^{a}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{t}>r)dr,\n\\end{align*}\nwhere $dS$ represents the surface measure on $\\partial D_{r}$.\nNow it follows from the scaling property of $Y^{(\\alpha)}$ and the change of variables $t^{-1\/\\alpha}r=s$ that\n\\begin{align*}\n&\\int_{0}^{a}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{t}>r)dr\n=\\int_{0}^{a}{\\mathbb P}(t^{1\/\\alpha}\\overline{Y}^{(\\alpha)}_{1}>r)dr\n=t^{1\/\\alpha}\\int_{0}^{t^{-1\/\\alpha}a}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>s)ds.\n\\end{align*}\nHence, we have\n$$\n\\frac{|D|-Q^{(\\alpha)}_{D}(t)}{t^{1\/\\alpha}}\\geq |\\partial D|\\left(\\frac{R-a}{R}\\right)^{d-1}\\int_{0}^{t^{-1\/\\alpha}a}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>s)ds\n$$\nand by taking $\\liminf$ we obtain\n$$\n\\liminf_{t\\to 0}\\frac{|D|-Q^{(\\alpha)}_{D}(t)}{t^{1\/\\alpha}}\\geq |\\partial D|\\left(\\frac{R-a}{R}\\right)^{d-1}{\\mathbb E}[\\overline{Y}^{(\\alpha)}_1].\n$$\nSince $a>0$ is arbitrary and \n${\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}]<\\infty$ \nwhen $\\alpha\\in (1,2)$, the conclusion of the theorem is true.\n{\\hfill $\\Box$ \\bigskip}\n\n\n\n\nNow we proceed to prove the opposite inequality of Lemma \\ref{lemma:lb}. \nFor a bounded $C^{1,1}$ open set $D$ satisfying the uniform \ninterior and exterior $R$-ball condition\nand $x\\in D$ with $\\delta_{D}(x)t\\}\\nonumber\\\\\n&\\subset&\\{\\tau_{H_{x}}^{(\\alpha)}\\leq t\\} \\cup \\{\\tau_{D}^{(\\alpha)}\\leq t <\\tau_{H_{x}}^{(\\alpha)}\\}.\n\\end{eqnarray}\nHence, we have\n\\begin{eqnarray}\\label{eqn:ub cases}\n&&|D|-Q^{(\\alpha)}_{D}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dx=\\int_{D_{R\/2}}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}\\leq t)dx +\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}\\leq t)dx\\nonumber\\\\\n&\\leq &\\int_{D_{R\/2}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dx+\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{H_{x}}\\leq t)dx +\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t<\\tau^{(\\alpha)}_{H_{x}})dx.\n\\end{eqnarray}\n\nWe deal with the first expression of \\eqref{eqn:ub cases} first.\nRecall the following facts from \\cite[(2.2)]{Val2017} and \\cite[Corollary 6.4]{BD89}:\n\\begin{equation}\\label{eqn:exp moment}\n{\\mathbb E}\\left[\\exp\\left(-\\frac{\\kappa^{2}}{S_{1}^{(\\alpha\/2)}}\\right)\\right]\\leq c_{\\alpha}\\kappa^{-\\alpha}\n\\end{equation}\nand\n\\begin{equation}\\label{eqn:BM exit}\n{\\mathbb P}\\left(\\tau^{(2)}_{B(0,R)}0\\}$.\n\\end{prop}\n\\noindent{\\bf Proof.} \nFor $\\delta_{D}(x)r\\}$, \nand this implies\n${\\mathbb P}_{(\\widetilde 0, r)}(\\tau^{(\\alpha)}_{H}\\leq 1)=\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>r)$. \nHence, we have\n$$\n\\int_{0}^{\\infty}{\\mathbb P}_{(\\widetilde 0, v)}(\\tau^{(\\alpha)}\\leq 1)dv\n=\\int_{0}^{\\infty}{\\mathbb P}(\n\\overline{Y}^{(\\alpha)}_{1}\\geq v)dv={\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}].\n$$\nBy \\cite[Corollary 2.1 (ii)]{Val2016}, ${\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}]<\\infty$ if $\\alpha\\in (1,2)$.\nHence, it follows from the Lebesgue dominated convergence theorem that\n$$\n\\lim_{t\\to 0}\\frac{\\int_{D\\setminus D_{R\/2}} {\\mathbb P}_{x}(\\tau^{(\\alpha)}_{H_{x}} \\leq t)dx}{t^{1\/\\alpha}}=|\\partial D|\\int_{0}^{\\infty}\n{\\mathbb P}_{(\\widetilde 0, r)}(\\tau^{(\\alpha)}_{H}\\leq 1)dr=\n|\\partial D|{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}].\n$$\n{\\hfill $\\Box$ \\bigskip}\n\n\n\nFinally, we estimate the last expression of \\eqref{eqn:ub cases}.\n\\begin{lemma}\\label{lemma:near boundary}\nLet $D\\subset {\\mathbb R}^{d}$, $d\\geq 2$, be a bounded $C^{1,1}$ \nopen set. \nThen, we have\n$$\n\\lim_{t\\to 0}\\frac{\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau_{D}^{(\\alpha)}\\leq t< \\tau^{(\\alpha)}_{H_{x}}\\right)dx}{t^{1\/\\alpha}}=0.\n$$\n\\end{lemma}\n\\noindent{\\bf Proof.} \nBy rotational invariance, we have\n\\begin{align}\\label{eqn:approx1}\n&\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau_{D}^{(\\alpha)}\\leq t< \\tau^{(\\alpha)}_{H_{x}}\\right)dx\n\\leq\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau^{(\\alpha)}_{B_{x}}\\leq t< \\tau^{(\\alpha)}_{H_{x}}\\right)dx\\nonumber\\\\\n&\\leq \\int_{0}^{R\/2}|\\partial D_{u}|{\\mathbb P}_{(\\widetilde 0, u)}\n\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, R), R)}\n\\leq t< \\tau^{(\\alpha)}_{H}\\right)du,\n\\end{align}\nwhere $H=\\{x=(x_{1},\\cdots ,x_{d}) : x_{d}>0\\}$.\nIt follows from \\cite[Lemma 6.7]{BD89} that for $uu, Y_{1}^{(\\alpha)}u, Y_{1}^{(\\alpha)}b, Y_{t}^{(\\alpha)}\\in (a,b))dx +\\int_{a}^{b}{\\mathbb P}_{x}(\\underline{Y}^{(\\alpha)}_{t}b, \\underline{Y}^{(\\alpha)}_{t}b, \\underline{Y}^{(\\alpha)}_{t}b, Y_{t}^{(\\alpha)}\\in (a,b))dx\\\\\n&=&t^{1\/\\alpha}\\int_{0}^{(b-a)t^{-1\/\\alpha}}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, -(b-a)t^{-1\/\\alpha}+u u, -(b-a)t^{-1\/\\alpha}+u u)\n$$\nand\n$$\n\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)du={\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}]<\\infty.\n$$\nHence, it follows from the Lebesgue dominated convergence theorem that the limit is\n\\begin{align*}\n&\\lim_{t\\to 0}\\frac{\\int_{a}^{b}{\\mathbb P}_{x}(\\tau_{(a,b)}^{(\\alpha)}\\leq t, Y_{t}^{(\\alpha)}\\in (a,b))dx}{t^{1\/\\alpha}}\\\\\n=&\\lim_{t\\to 0}\\frac{t^{1\/\\alpha}\\int_{0}^{(b-a)t^{-1\/\\alpha}}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, -(b-a)t^{-1\/\\alpha}+u u, Y_{1}^{(\\alpha)}0$ and $\\alpha\\in (0,2)$, we have\n$$\n\\lim_{t\\to 0}\\frac{{\\mathbb E}[\\left(S_{1}^{(\\alpha\/2)}\\right)^{\\alpha\/2}, 0u, Y_{1}^{(\\alpha)}T_{t}^{(\\beta\/2)})dx.\n$$\n\nSince $\\{\\tau_{D}^{(1)} \\leq t\\}\\subset \\{\\tau_{D}^{(\\alpha)} \\leq T_{t}^{(\\beta\/2)}\\}$, we have $|D|-Q_{D}^{(1)}(t)\\leq |D|-\\widetilde{Q}^{(\\alpha,\\beta)}_{D}(t)$. \nWe will show that \n\\begin{equation}\\label{eqn:Cauchy aux7}\n\\limsup_{t\\to 0}\\frac{|D|-\\widetilde{Q}^{(\\alpha,\\beta)}_{D}(t)}{t\\ln(1\/t)}\\leq \n\\frac{|\\partial D|}{\\pi} +\n\\frac{\\int_{0}^{\\infty}\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}0$,\nthere exists $\\delta=\\delta(\\varepsilon)$ such that for all $u\\leq \\delta$,\n\\begin{eqnarray*}\n\\frac{|D|-Q_{D}^{(\\alpha)}(u)}{u^{1\/\\alpha}} \n\\leq \\left(\\frac{\\Gamma(1-\\frac{1}{\\alpha})}{\\pi} + \\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}u, Y_{1}^{(\\alpha)}0$ is arbitrary, this establishes \\eqref{eqn:Cauchy aux7}.\n{\\hfill $\\Box$ \\bigskip}\n\n\nWe now recall the definition for the double gamma function from \\cite[(4.4)]{Ku11}. \nFor $z\\in \\mathbb{C}$ and $\\tau \\in \\mathbb{C}$ with $|\\text{arg}(\\tau)|<\\pi$, we define \n$$\nG(z;\\tau)=\\frac{z}{\\tau}e^{\\frac{az}{\\tau} + \\frac{bz^2}{2\\tau}}\n{\\textstyle \\prod}_{m\\geq 0}{\\textstyle \\prod}^{'}_{n\\geq 0}\\left( 1+\\frac{z}{m\\tau +n}\\right)\\exp\\left(-\\frac{z}{m\\tau +n} +\\frac{z^2}{2(m\\tau +n)^2}\\right),\n$$\nwhere the prime in the second product means that the term corresponding to $m=n=0$ is omitted. \n\n\\begin{lemma}\\label{lemma:double gamma bounded}\nLet $K$ be a compact set in $\\mathbb{C}$. \nThen there exists a constant $c=c(K)>0$ such that \n$$\n\\left| G(z;\\tau) \\right| \\leq c \\text{ for all } z\\in K \\text{ and } \\tau\\in [1,2].\n$$\n\\end{lemma}\n\\noindent{\\bf Proof.} \nOn the compact set $K\\times [1, 2]$, $\\frac{z}{\\tau}e^{\\frac{az}{\\tau} + \\frac{bz^2}{2\\tau}}$ is continuous and hence bounded. \nWe only need to prove that the double infinite product in $G(z, \\tau)$ is bounded. \nRecall the canonical factor $E_{2}(z)$ (see \\cite[p. 145]{SS complex}):\n$$\nE_{2}(z)=(1-z)e^{z+\\frac{z^2}{2}}.\n$$\nThe double infinite product in $G(z;\\tau)$ can be written as\n$$\n {\\textstyle \\prod}_{m\\geq 0}{\\textstyle \\prod}^{'}_{n\\geq 0}\\left( 1+\\frac{z}{m\\tau +n}\\right)\\exp\\left(-\\frac{z}{m\\tau +n} +\\frac{z^2}{2(m\\tau +n)^2}\\right)\n = {\\textstyle \\prod}_{m\\geq 0}{\\textstyle \\prod}^{'}_{n\\geq 0}E_{2}(-\\frac{z}{m\\tau +n}).\n$$\nIt follows from \\cite[Lemma 4.2]{SS complex} for $|w|\\leq 1\/2$, $|1-E_{2}(w)|\\leq c_{1}|w|^{3}$ for some constant $c_{1}$.\nSince $|-\\frac{z}{m\\tau +n}| \\leq \\frac{|z|}{m+n}$, \nall but finitely many terms of the form $|-\\frac{z}{m\\tau +n}|$ in the double infinite product are less than $\\frac12$, and since each of these finitely many terms are bounded on $K\\times [1, 2]$, we may assume $|-\\frac{z}{m\\tau +n}|\\leq \\frac12$ on $K\\times [1, 2]$ for all $n,m$.\n\nNow it follows from \\cite[Lemma 4.2]{SS complex} for all \n$(z, \\tau)\\in K\\times [1, 2]$ that\n$$\n\\left| 1-E_{2}(-\\frac{z}{m\\tau +n})\\right| \\leq c_{1}\\left |\\frac{z}{m\\tau +n}\\right|^{3}\\leq \\frac{c_{2}}{(m+n)^{3}} \\text{ for all } n,m.\n$$\nIt is easy to see that \n$$\n\\sum_{m\\geq 0, n\\geq 0, (m,n)\\neq (0, 0)}\\frac{1}{(m+n)^{3}}<\\infty.\n$$\nNote that for any $|w|\\leq \\frac{1}{2}$ we have $\\ln |1-w|\\leq 2|w|$.\nThere exist positive integers $N_1$ and $M_1$ such that $|1-E_{2}(-\\frac{z}{m\\tau +n})|\\leq \\frac{1}{2}$ on $K\\times [1, 2]$ for all $n\\geq N_{1}$ and $m\\geq M_{1}$.\nHence, for any $M, N \\in {\\mathbb N}$ large we have \n\\begin{align*}\n&\\left|{\\textstyle \\prod}_{M_{1}\\leq m \\leq M}{\\textstyle \\prod}_{N_{1}\\leq n\\leq N} E_{2}(-\\frac{z}{m\\tau +n}) \\right|\n={\\textstyle \\prod}_{M_{1}\\leq m \\leq M}{\\textstyle \\prod}_{N_{1}\\leq n\\leq N}\\left|1- (1- E_{2}(-\\frac{z}{m\\tau +n}) )\\right|\\\\\n&={\\textstyle \\prod}_{M_{1}\\leq m \\leq M}{\\textstyle \\prod}_{N_{1}\\leq n\\leq N}\\exp \\left( \\ln \\left| (1- (1- E_{2}(-\\frac{z}{m\\tau +n}) )) \\right| \\right)\\\\\n&= \\exp \\left( \\sum_{M_{1}\\leq m \\leq M,N_{1}\\leq n\\leq N} \\ln \\left|(1- (1- E_{2}(-\\frac{z}{m\\tau +n}) )) \\right| \\right)\\\\\n&\\leq \\exp \\left( 2\\sum_{M_{1}\\leq m \\leq M,N_{1}\\leq n\\leq N} \\left| 1- E_{2}(-\\frac{z}{m\\tau +n}) \\right| \\right)\\leq \\exp \\left( 2c_2\\sum_{M_{1}\\leq m \\leq M,N_{1}\\leq n\\leq N} \\frac{1}{(m+n)^{3}} \\right)\\\\\n&\\leq \\exp \\left( 2c_2\\sum_{M_{1}\\leq m, N_{1}\\leq n} \\frac{1}{(m+n)^{3}} \\right).\n\\end{align*}\nBy letting $M, N\\to \\infty$ we see that the double infinite product is bounded on \n$K\\times [1, 2]$.\n{\\hfill $\\Box$ \\bigskip}\n\nIt follows from \\cite[Proposition VIII.1-4]{Ber} that there exists a constant $C>0$ such that\n\\begin{equation}\\label{eqn:same asymptotic}\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_1>u)\\sim{\\mathbb P}(Y^{(\\alpha)}_1>u)\\sim Cu^{-\\alpha} \n\\text{ as } u\\to \\infty.\n\\end{equation}\nLet $\\rho:={\\mathbb P}(Y_1^{(\\alpha)}>0)$. We say $Y^{(\\alpha)}\\in C_{k,l}$ if\n\\begin{align}\\label{eqn:ckl}\n&(\\alpha, \\rho)\\in \\{\\alpha\\in (0,1), \\rho\\in(0,1)\\} \\cup \\{\\alpha=1, \\rho=1\/2\\}\\cup \\{\\alpha\\in (1,2), \\rho\\in [1-\\frac{1}{\\alpha}, \\frac{1}{\\alpha}]\\} \\text{ and}\\nonumber\\\\\n&\\frac{1}{2}+k=\\frac{l}{\\alpha}, \\text{ or equivalently } \\alpha=\\frac{2l}{1+2k} \\text{ for some } k,l\\in {\\mathbb N}.\n\\end{align}\nNote that this condition already appeared in \\cite[\\text{Definition 1}]{Ku11}. \n\n\\begin{lemma}\\label{lemma:uniform ub}\nSuppose that $Y^{(\\alpha)}\\in C_{k,k+1}$\nfor some $k\\in \\mathbb{N}$.\nThen $\\overline{Y}_1^{(\\alpha)}$ has a density $\\overline{p}^{(\\alpha)}(x)$ and there exists \na constant $A>0$, \nindependent of all $\\alpha\\in(1,2)$ satisfying $Y_{t}^{(\\alpha)}\\in C_{k,k+1}$, such that\n$$\n\\overline{p}^{(\\alpha)}(x)\\leq C\\alpha x^{-1-\\alpha} +Ax^{-3}\n$$\nfor all $x>0$, where $C$ is the constant in \\eqref{eqn:same asymptotic}.\n\\end{lemma}\n\\noindent{\\bf Proof.} \nThe proof is similar to that of \\cite[Theorem 9]{Ku11} with a focus on establishing \na uniform constant $A$. \nIt follows from the proof of \\cite[Theorem 9]{Ku11} that $\\overline{Y}_1^{(\\alpha)}$ has a density $\\overline{p}^{(\\alpha)}(x)$ given by the inverse Mellin transform\n$$\n\\overline{p}^{(\\alpha)}(x)=\\frac{1}{2\\pi i}\\int_{1+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds,\n$$\nwhere $M(s,\\alpha)$ is the Mellin transform given by\n$$\nM(s,\\alpha)={\\mathbb E}[(\\overline{Y}^{(\\alpha)}_{1})^{s-1}], \\quad s\\in \\mathbb{C}.\n$$\nWe remark here that we write the Mellin transform as $M(s,\\alpha)$ instead of $M(s)$ to emphasize its dependence on $\\alpha$.\nIt follows from \\cite[Lemma 2]{Ku11} that $M(s,\\alpha)$ can be extended to a meromorphic function on $\\mathbb{C}$ whose simples poles are at $s_{m,n}:=m+\\alpha n$, where $m\\leq 1-(k+1)$ and $n\\in \\{0,1,\\cdots, k\\}$, or $m\\geq 1$ and $n\\in \\{1,2,\\cdots, k\\}$ with residues\n$$\n\\text{Res}(M(s,\\alpha), s_{m,n})=c_{m-1,n}^{+},\n$$\nwhere $c^{+}_{m-1,n}$ is the constant defined in \\cite[(7.5)]{Ku11}.\n\nSince $Y^{(\\alpha)}\\in C_{k,k+1}$, \nit follows from \\eqref{eqn:ckl} that $\\alpha=\\frac{2+2k}{1+2k}\\in (1,2)$.\nIf $m\\geq 1$ and $n\\in \\{1,\\cdots, k\\}$ then $s_{m,n}=m+\\alpha n$ has a nonempty intersection with $[1,3]$ if and only if $m=n=1$. In particular, $s_{1,1}=1+\\alpha$. \nIf $m\\leq 1-(k+1)=-k$ and $n\\in \\{0,\\cdots, k\\}$, we have \n$$\ns_{m,n}=m+\\alpha n \\leq -k +\\frac{2+2k}{1+2k}k=\\frac{k}{1+2k} <\\frac{1}{2}.\n$$\nHence, the only simple pole of $M(s,\\alpha)$ with $\\text{Re}(s)\\in [1,3]$ is $s_{1,1}=1+\\alpha$.\n\n\nWe note here that although it is written as $y\\to \\infty$ in \\cite[Lemma 3]{Ku11}, it is actually true for all $|y|\\to \\infty$ by checking the proof there as \\cite[(7.15)]{Ku11} is true for all $z\\to\\infty$ with $|\\text{arg}(z)|<\\pi$.\nHence, by taking a rectangle with vertices at $1\\pm Pi$ and $3\\pm Pi$ with the help of the residue theorem, then letting $P\\to \\infty$ and using \\cite[Lemma 3]{Ku11}, we get\n\\begin{eqnarray*}\n&&\\overline{p}^{(\\alpha)}(x)=\\frac{1}{2\\pi i}\\int_{1+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds \\\\\n&=&\\text{Res}(M(s,\\alpha), s_{1,1}^{+})x^{-1-\\alpha} +\\frac{1}{2\\pi i}\\int_{3+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds\\\\\n&=&c_{0,1}^{+}x^{-1-\\alpha}+\\frac{1}{2\\pi i}\\int_{3+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds,\n\\end{eqnarray*}\nwhere $\\text{Res}(M(s,\\alpha), s_{1,1}^{+})$ is the residue of $M(s,\\alpha)$ at $s_{1,1}^{+}$.\n\nWe claim that the constant $c_{0,1}^{+}$ must be $C\\alpha$, where $C$ is from \\eqref{eqn:same asymptotic} as we will show that the reminder is $O(x^{-3})$, which will in turn imply that there exist constants $c_{1}, c_{2}\\in {\\mathbb R}$ such that \n$$\nc_{0,1}^{+}x^{-1-\\alpha} +c_1x^{-3} \\leq \\overline{p}^{(\\alpha)}(x) \\leq c_{0,1}^{+}x^{-1-\\alpha} +c_2x^{-3}\n$$\nfor all sufficiently large $x$.\nBy integrating on $(u,\\infty)$ we obtain \n$$\n\\frac{c_{0,1}^{+}}{\\alpha}u^{-\\alpha} +\\frac{c_1}{2} u^{-2}\\leq \n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u) \n\\leq \\frac{c_{0,1}^{+}}{\\alpha}u^{-\\alpha} +\\frac{c_2}{2}u^{-2},\n$$\nfor all sufficiently large $u>0$.\nComparing the equation above with \\eqref{eqn:same asymptotic} we conclude that \n$\\alpha^{-1}c_{0,1}^{+}=C$.\nThis shows that the leading term of $\\overline{p}^{(\\alpha)}(x)$ is $C\\alpha x^{-1-\\alpha}$.\n\nHence, the proof will be complete once we show that \n\\begin{equation}\\label{eqn:Mellin error}\n\\left| \\frac{1}{2\\pi i}\\int_{3+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds\\right| \n\\leq Ax^{-3},\n\\end{equation}\nwhere the constant $A$ is independent of all $\\alpha\\in (1,2)$ satisfying \n$Y^{(\\alpha)}\\in C_{k,k+1}$.\n\nFrom \\cite[Lemma 3]{Ku11} we know that \n\\begin{equation}\\label{eqn:Mellin}\n\\ln|M(3+iy,\\alpha)|=-\\frac{\\pi |y|}{2\\alpha} +o(y) \\text{ as } |y|\\to\\infty.\n\\end{equation}\nThe proof of \\eqref{eqn:Mellin} in \\cite[Lemma 3]{Ku11} depends on \\cite[(4.5)]{BK97}, which is a uniform estimate of the double gamma functions. Hence, the error term in \\eqref{eqn:Mellin} is uniform for all $\\alpha\\in (1,2)$ and \nthere exists $N>0$, independent of $\\alpha\\in (1,2)$, such that \n$$\n\\ln|M(3+iy,\\alpha)|\\leq -\\frac{\\pi |y|}{5} \\text{ for all } |y|\\geq N \\text{ and } \\alpha\\in(1,2).\n$$\nHence, we have \n\\begin{equation}\\label{eqn:Mellin ub}\n|M(3+iy, \\alpha)|\\leq e^{-\\frac{\\pi|y|}{5}} \\text{ for all } |y|\\geq N \\text{ and } \\alpha\\in(1,2).\n\\end{equation}\nIt follows from \\cite[Theorem 8]{Ku11} that the Mellin transform $M(s,\\alpha)$ can be written as\n$$\nM(s,\\alpha)=\\alpha^{s-1}\\frac{G(\\alpha\/2;\\alpha)}{G(\\alpha\/2+1;\\alpha)}\\frac{G(\\alpha\/2+2-s;\\alpha)}{G(\\alpha\/2-1+s;\\alpha)}\\frac{G(\\alpha-1+s;\\alpha)}{G(\\alpha+1-s;\\alpha)},\n$$\nwhere $G(z;\\tau)$ is the double gamma function.\n$G(z;\\tau)$ has simple zeroes on the lattice $m\\tau+n$, $m,n\\leq 0$.\nBy a simple calculation one can check that the double gamma functions in the denominators above\nhave no zero with $s=3+iy$ for $|y|\\leq N$.\nIt follows from Lemma \\ref{lemma:double gamma bounded} that there exists a constant $c>0$ such that\n$$\n|M(3+iy,\\alpha)|\\leq ce^{-\\frac{\\pi|y|}{5}} \\text{ for all } |y|\\leq N \\text{ and } \\alpha \\in (1,2),\n$$\nand combining with \\eqref{eqn:Mellin ub} we have \n$$\n|M(3+iy,\\alpha)|\\leq ce^{-\\frac{\\pi|y|}{5}} \\text{ for all } y\\in {\\mathbb R} \\text{ and } \\alpha \\in (1,2).\n$$\n\nHence, \\eqref{eqn:Mellin error} can be estimated as \n$$\n\\left| \\frac{1}{2\\pi i}\\int_{3+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds\\right| \\leq \\frac{x^{-3}}{2\\pi}\\int_{-\\infty}^{\\infty}|M(3+it, \\alpha)|dt\\leq\\frac{x^{-3}}{2\\pi}\\int_{-\\infty}^{\\infty}ce^{-\\frac{\\pi |t|}{5}}dt:=\nAx^{-3}.\n$$\n{\\hfill $\\Box$ \\bigskip}\n\nLet $p^{(\\alpha)}(x)$ be the transition density of $Y^{(\\alpha)}_1$.\nThe following lemma is a variation of \\cite[Propositions 7.1.1 and 7.1.2]{Ko} with an explicit error estimate. \n\\begin{lemma}\\label{lemma:uniform lb}\nLet $\\alpha\\in(1,2)$ and $n\\in{\\mathbb N}$. Then, we have \n$$\np^{(\\alpha)}(x)=\n\\frac{1}{\\pi}\\sum_{k=1}^{n}\\frac{(-1)^{k+1}\\Gamma(1+k\\alpha)\\sin(\\frac{k\\alpha\\pi}{2}) }{k!}x^{-1-\\alpha k}+E(x,\\alpha),\n$$\nwhere \n$$\n\\left|E(x,\\alpha)\\right|\\leq \\frac{2\\Gamma(2n+2)}{\\pi n!}x^{-2-n} \\text{ for all } x\\geq 1 \\text{ and } \\alpha \\in(1,2).\n$$\nIn particular, for any $x\\geq 1$ and $\\alpha\\in (1,2)$ we have\n$$\n \\frac{\\Gamma(1+\\alpha)\\sin\\frac{\\pi\\alpha}{2}}{\\pi}x^{-1-\\alpha}-\\frac{12}{\\pi}x^{-3} \\leq p^{(\\alpha)}(x)\\leq \\frac{\\Gamma(1+\\alpha)\\sin\\frac{\\pi\\alpha}{2}}{\\pi}x^{-1-\\alpha}+\\frac{12}{\\pi}x^{-3},\n$$\nand in fact \n$$\n \\frac{\\Gamma(1+\\alpha)\\sin\\frac{\\pi\\alpha}{2}}{\\pi}=C\\alpha,\n$$\nwhere $C$ is the constant in \\eqref{eqn:same asymptotic}.\n\\end{lemma}\n\\noindent{\\bf Proof.} \nTo prove this lemma, we consider a 1-dimensional $\\alpha$-stable processes $Y^{(\\alpha), \\gamma}_{t}$ with\n$$\n{\\mathbb E}[e^{iy Y_{t}^{(\\alpha), \\gamma}}]=\\exp(-t|y|^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}\\text{sgn}(y)}),\n$$\nwhere $\\gamma$ represents the skewness of $Y^{(\\alpha), \\gamma}_{t}$ and $\\text{sgn}(y):=1_{\\{y\\geq 0\\}}- 1_{\\{y<0\\}}$. \nWe use \n$p^{(\\alpha)}(x, \\gamma)$\nto denote the density of $Y^{(\\alpha), \\gamma}_{1}$. When $\\gamma=0$, $Y^{(\\alpha), \\gamma}_{t}$ reduces to $Y^{(\\alpha)}_{t}$.\nBy the Fourier inversion we have \n\\begin{eqnarray*}\n&&p^{(\\alpha)}(x,\\gamma)\n=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{-ixy}\\exp(-|y|^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}\\text{sgn}(y)})dy\\\\\n&=&\\frac{1}{2\\pi}\\int_{0}^{\\infty}e^{-ixy}\\exp(-|y|^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}})dy+\\frac{1}{2\\pi}\\int_{-\\infty}^{0}e^{-ixy}\\exp(-|y|^{\\alpha}e^{-\\frac{i\\pi\\gamma}{2}})dy\\\\\n&=&\\frac{1}{2\\pi}\\int_{0}^{\\infty}e^{-ixy}\\exp(-|y|^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}})dy+\\frac{1}{2\\pi}\\int^{\\infty}_{0}e^{ixy}\\exp(-|y|^{\\alpha}e^{-\\frac{i\\pi\\gamma}{2}})dy\\\\\n&=&\\frac{1}{\\pi}\\text{Re}\\int_{0}^{\\infty}e^{-ixy}\\exp(-y^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}})dy,\n\\end{eqnarray*} \nwhere $\\text{Re}(\\cdot)$ represents the real part of the argument. \nBy the Taylor expansion with remainder we can write $e^{-ixy}$ as\n$$\ne^{-ixy}=\\sum_{k=0}^{n-1}\\frac{(-ix)^{k}}{k!}y^{k} +\\frac{x^{n}y^{n}}{n!}R,\n$$\nwhere $|R|\\leq 1$.\nSince, \n$$\n\\int_{0}^{\\infty}y^{\\beta}\\exp(-\\lambda y^{\\alpha})dy=\\alpha^{-1}\\lambda^{-\\frac{\\beta+1}{\\alpha}}\\Gamma(\\frac{\\beta+1}{\\alpha}),\n$$\nwe have \n\\begin{align}\\label{eqn:HK}\n&p^{(\\alpha)}(x,\\gamma)\n=\\frac{1}{\\pi}\\text{Re}\\int_{0}^{\\infty}\\left( \\sum_{k=0}^{n-1}\\frac{(-ix)^{k}}{k!}y^{k} +\\frac{x^{n}y^{n}}{n!}R\\right)\\exp(-y^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}})dy\\nonumber\\\\\n&=\\frac{1}{\\pi}\\text{Re}\\sum_{k=0}^{n-1}\\frac{(-ix)^{k}}{k!}\\frac{1}{\\alpha}(e^{\\frac{i\\gamma\\pi}{2}})^{-\\frac{k+1}{\\alpha}}\\Gamma(\\frac{k+1}{\\alpha})+\n\\text{Re}\\left(\\frac{x^{n}}{\\pi\\alpha n!}\\Gamma(\\frac{n+1}{\\alpha})(e^{\\frac{i\\gamma\\pi}{2}})^{-\\frac{n+1}{\\alpha}}R\\right)\\nonumber\\\\\n&=\\frac{1}{\\alpha\\pi}\\text{Re}\\sum_{k=1}^{n}\\frac{(-ix)^{k-1}}{(k-1)!}e^{-\\frac{i\\gamma\\pi k}{2\\alpha}}\\Gamma(\\frac{k}{\\alpha})\n+\\text{Re}\\left(\\frac{x^n}{\\pi\\alpha n!}\\Gamma(\\frac{n+1}{\\alpha})e^{-\\frac{i\\gamma\\pi(n+1)}{2\\alpha}}R\\right)\\nonumber\\\\\n&=\\frac{-1}{\\alpha\\pi x}\\text{Re}\\sum_{k=1}^{n}\\frac{(-x)^{k}}{(k-1)!}\\exp\\left(-i(\\frac{k\\pi(\\gamma-\\alpha)}{2\\alpha} +\\frac{\\pi}{2})\\right)\\Gamma(\\frac{k}{\\alpha})+\\text{Re}\\left(\\frac{x^n}{\\pi\\alpha n!}\\Gamma(\\frac{n+1}{\\alpha})e^{-\\frac{i\\gamma\\pi(n+1)}{2\\alpha}}R\\right)\\nonumber\\\\\n&:=\\frac{1}{\\pi x}\\sum_{k=1}^{n}\\frac{(-x)^{k}}{k!}\\sin(\\frac{k\\pi(\\gamma-\\alpha)}{2\\alpha})\\Gamma(1+\\frac{k}{\\alpha})+R(x,\\alpha),\n\\end{align}\nwhere \n\\begin{equation}\\label{eqn:remainder}\n\\left| R(x,\\alpha)\\right|\\leq \\frac{x^n}{\\pi\\alpha n!}\\Gamma(\\frac{n+1}{\\alpha}).\n\\end{equation}\n\nIt follows from Zolotarev's identity (see \\cite[(7.10)]{Ko}) for $\\alpha\\in(\\frac12,1)\\cup (1,2)$,\n$$\np^{(\\alpha)}(x,\\gamma)=x^{-1-\\alpha}p^{(1\/\\alpha)}(x^{-\\alpha}, \\frac{\\gamma+1}{\\alpha}-1).\n$$\nHence, from \\eqref{eqn:HK} we have \n\\begin{eqnarray*}\n&&p^{(\\alpha)}(x,\\gamma)=x^{-1-\\alpha}p^{(1\/\\alpha)}(x^{-\\alpha}, \\frac{\\gamma+1}{\\alpha}-1)\\\\\n&=&x^{-1-\\alpha}\\left(\\frac{1}{\\pi x^{-\\alpha}}\\sum_{k=1}^{n}\\frac{(-x^{-\\alpha})^{k}}{k!}\\sin(\\frac{k\\pi(\\gamma-\\alpha)}{2})\\Gamma(1+k\\alpha) +R(x^{-\\alpha}, \\frac{1}{\\alpha})\\right).\n\\end{eqnarray*}\nNow let $\\gamma=0$ and define $E(x,\\alpha):=x^{-1-\\alpha}R(x^{-\\alpha}, \\frac{1}{\\alpha})$. From \\eqref{eqn:remainder} we have \n$$\n\\left|E(x,\\alpha)\\right|\\leq \\frac{\\alpha x^{-(n+1)\\alpha-1}}{\\pi n!}\\Gamma(\\alpha(n+1)).\n$$\nHence, for all $\\alpha\\in (1,2)$ and $x\\geq 1$ we have \n$$\n\\left|E(x,\\alpha)\\right|\\leq \\frac{2 x^{-n-2}}{\\pi n!}\\Gamma(2(n+1)).\n$$\n\nThe fact $\\frac{\\Gamma(1+\\alpha)\\sin\\frac{\\pi\\alpha}{2}}{\\pi}=C\\alpha$ can be proved using a similar argument as in the proof of Lemma \\ref{lemma:uniform ub}.\n{\\hfill $\\Box$ \\bigskip}\n\n\n\\begin{lemma}\\label{lemma:Cauchy cancel}\nLet $\\alpha\\in (1,2)$ and suppose that \n$Y^{(\\alpha)}\\in C_{k,k+1}$ for some $k\\in \\mathbb{N}$. \nThere exists a function $\\phi$, independent of $\\alpha\\in (1,2)$, such that \n$$\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_1>u, Y^{(\\alpha)}_1\\leq u)\\leq \\phi(u) \\text{ with } \\int_{0}^{\\infty}\\phi(u)du<\\infty.\n$$\n\\end{lemma}\n\\noindent{\\bf Proof.} \nNote that \n$$\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}\\leq u)={\\mathbb P}(\\overline{Y}^{(\\alpha)}_1>u)-{\\mathbb P}(Y_{1}^{(\\alpha)}>u).\n$$\nIt follows from Lemmas \\ref{lemma:uniform ub} and \\ref{lemma:uniform lb} that\n$$\n\\overline{p}^{(\\alpha)}(x)\\leq C\\alpha x^{-1-\\alpha} +\nAx^{-3} \\text{ and } \np^{(\\alpha)}(x)\\geq C\\alpha x^{-1-\\alpha} -\\frac{12}{\\pi}x^{-3},\n$$\nwhere $A$ is independent of $\\alpha\\in (1,2)$.\n\nHence, we have \n\\begin{eqnarray*}\n&&{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}\\leq u)={\\mathbb P}(\\overline{Y}^{(\\alpha)}_1>u)-{\\mathbb P}(Y_{1}^{(\\alpha)}>u)\\\\\n&\\leq&\\int_{u}^{\\infty}\\left(C\\alpha x^{-1-\\alpha} +\nAx^{-3}-C\\alpha x^{-1-\\alpha} +\\frac{12}{\\pi}x^{-3} \\right)dx=\\frac{A}{2}u^{-2}+\\frac{6}{\\pi}u^{-2}.\n\\end{eqnarray*}\nFinally, we set \n$$\n\\phi(u):=1_{\\{01\\}}\n(\\frac{A}{2}+\\frac{6}{\\pi})u^{-2}.\n$$\n{\\hfill $\\Box$ \\bigskip}\n\n\\begin{prop}\\label{prop:Cauchy ub}\nLet $D$ be a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$, $d\\geq 2$. Then, we have\n$$\n\\limsup_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}\\leq \\frac{|\\partial D|}{\\pi}.\n$$\n\\end{prop}\n\\noindent{\\bf Proof.} \nFrom Lemma \\ref{lemma:Cauchy ub} for any $\\alpha\\in(1,2)$ we have \n\\begin{equation}\\label{eqn:Cauchy ub main}\n\\limsup_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}\\leq \\frac{|\\partial D|}{\\pi} +\n\\frac{\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}1.\n\\end{equation}\nNote that from \\eqref{eqn:ckl} for \n$Y^{(\\alpha)}\\in C_{k,k+1}$\nwe have $\\alpha \\in (1,2)$ and as $k\\to \\infty$, $\\alpha\\downarrow 1$.\nHence, it follows from Lemma \\ref{lemma:Cauchy cancel} and the Lebesgue dominated convergence theorem that\n$$\n\\lim_{Y_{t}^{(\\alpha)}\\in C_{k,k+1}, k\\to \\infty}\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}\\leq u)du\n=\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u, Y_{1}^{(1)}\\leq u)du.\n$$\nNote that the density $p^{(1)}(x)$ of $Y^{(1)}_{1}$ is given by \n$$\np^{(1)}(x)=\\frac{1}{\\pi(1+x^2)}\n$$\nand $\\int_{u}^{\\infty}{\\mathbb P}(Y_{1}^{(1)}>u)du=\\int_{u}^{\\infty}\\frac{1}{\\pi (1+x^2)}dx=\\frac{1}{\\pi}(\\frac{\\pi}{2}-\\arctan(u))=\\frac{1}{\\pi}\\arctan(\\frac{1}{u})$.\nBy the Taylor expansion of $\\arctan x =\\sum_{n=1}^{\\infty}(-1)^{n+1}\\frac{x^{2n-1}}{2n-1}$ for $|x|<1$ we have $\\frac{1}{\\pi}\\arctan(1\/u)\\geq \\frac{1}{\\pi u} -\\frac{1}{2\\pi u^3}$ for all sufficiently large $u$.\nHence, it follows from \\cite[(3.5)]{P20} that \n\\begin{eqnarray*}\n&&{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u, Y_{1}^{(1)}\\leq u ) ={\\mathbb P}(\\overline{Y}^{(1)}_{1}>u)-{\\mathbb P}(Y_{1}^{(1)}> u)\\\\\n&=&{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u)-\\frac{\\arctan(1\/u)}{\\pi}\\leq{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u)-\\frac{1}{\\pi u} +\\frac{1}{2\\pi u^3}\\leq \\frac{4}{\\pi^2}\\frac{\\ln u}{u^3} +\\frac{1}{2\\pi u^3}\n\\end{eqnarray*}\nfor all sufficiently large $u>0$, and this implies \n\\begin{equation}\\label{eqn:Cauchy finite}\n\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u, Y_{1}^{(1)}\\leq u)du<\\infty.\n\\end{equation}\n\nHence, if follows from \\eqref{eqn:Gamma} and \\eqref{eqn:Cauchy finite} that\n$$\n\\lim_{Y^{(\\alpha)}\\in C_{k,k+1}, k\\to \\infty}\n\\frac{\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}\\tilde{T}$ and possibly also two\nadditional dynamical variables---which are characterized by the following\ntwo properties:\n\n(i) For the same variables of the original dynamical system $D$ the new\ndynamical system $\\tilde{D}$ yields, over the time interval $\\tilde{T}$,\nhence over an \\textit{arbitrarily long} time, a dynamical evolution which\nmimics \\textit{arbitrarily closely} that yielded by the original system $D$;\nup to corrections of order $t\/\\tilde{T}$, or possibly even \\textit\nidentically}.\n\n(ii) The system $\\tilde{D}$ is \\textit{isochronous}: for arbitrary initial\ndata all its solutions are completely periodic with a period $T$, which can\nalso be arbitrarily assigned, except for the\\ (obviously necessary)\ncondition $T>\\tilde{T}$.\n\nMoreover it has been shown \\cite{CL2007,calogero} that, if the dynamical\nsystem $D$ is a many-body problem characterized by a (standard, autonomous)\nHamiltonian $H$ which is translation-invariant (i. e., it features no\nexternal forces), other (also autonomous) Hamiltonians $\\tilde{H}$\ncharacterizing \\textit{modified} many-body problems can be manufactured\nwhich feature the \\textit{same} dynamical variables as $H$ (i. e., in this\ncase there is no need to introduce two additional dynamical variables) and\nwhich yield a time evolution \\textit{quite close}, or even \\textit{identical\n, to that yielded by the original Hamiltonian $H$ over the \\textit\narbitrarily assigned} time $\\tilde{T}$, while being \\textit{isochronous}\nwith the \\textit{arbitrarily assigned} period $T$, of course with $T>\\tilde{\n}$.\n\nLet us emphasize that the class of Hamiltonians $H$ for which this result is\nvalid is quite general. In particular it includes the standard Hamiltonian\nsystem describing an \\textit{arbitrary} number $N$ of point particles with\n\\textit{arbitrary} masses moving in a space of \\textit{arbitrary} dimensions\n$d$ and interacting among themselves via potentials depending \\textit\narbitrarily} from the interparticle distances (including the possibility of\nmultiparticle forces), being therefore generally valid for any realistic\nmany-body problem, hence encompassing most of nonrelativistic physics. This\nresult is moreover true, \\textit{mutatis mutandis}, in a quantal context.\n\nFor instance let us tersely review here the case of the standard Hamiltonian\ndescribing the many-body problem, but focusing---merely for notational\nsimplicity---on the case with equal particles (and setting their mass to\nunity) and a one-dimensional setting, which reads (in self-evident standard\nnotation)\n\\begin{subequations}\n\\begin{equation}\nH\\left( \\underline{p},\\underline{q}\\right) =\\frac{1}{2}\\sum_{n=1}^{N}\\left(\np_{n}^{2}\\right) +V\\left( \\underline{q}\\right) \\label{H}\n\\end{equation\nwith the potential being translation-invariant, $V\\left( \\underline{q\n+a\\right) =V\\left( \\underline{q}\\right) $ (i. e., no external forces) but\notherwise unrestricted (except for the condition---again, for\nsimplicity---that the time-evolution yielded by this Hamiltonian be\nnonsingular). The standard approach to treat this problem is to introduce\nthe center-of-mass coordinate $Q\\left( \\underline{q}\\right) =\\frac{1}{N\n\\sum_{n=1}^{N}q_{n}$ and the total momentum $P\\left( \\underline{p}\\right)\n=\\sum_{n=1}^{N}p_{n},$ and to then focus on the coordinate and momenta in\nthe center of mass, $x_{n}=q_{n}-Q\\left( \\underline{q}\\right) ,$ \ny_{n}=p_{n}-P\\left( \\underline{p}\\right) \/N$, which characterize the physics\nof the problem and whose evolution is determined by the Hamiltonian $h\\left(\n\\underline{y},\\underline{x}\\right) $ defined as follows\n\\begin{equation}\nH\\left( \\underline{p},\\underline{q}\\right) =\\frac{\\left[ P\\left( \\underline{\n}\\right) \\right] ^{2}}{2N}+h\\left( \\underline{y},\\underline{x}\\right) ~,\n\\label{Hh}\n\\end{equation\n\\begin{equation}\nh\\left( \\underline{y},\\underline{x}\\right) =\\frac{1}{2}\\sum_{n=1}^{N}\\left(\ny_{n}^{2}\\right) +V\\left( \\underline{x}\\right) ~, \\label{h}\n\\end{equation\nwhere $V\\left( \\underline{x}\\right) =V\\left( \\underline{q}\\right) $ thanks\nto the assumed translation-invariance of $V\\left( \\underline{q}\\right) $. An\n\\textit{isochronous} Hamiltonian $\\tilde{H}\\left( \\underline{p},\\underline{q\n;T\\right) $ reads then as follows,\n\\end{subequations}\n\\begin{subequations}\n\\begin{equation}\n\\tilde{H}\\left( \\underline{p},\\underline{q};T\\right) =\\frac{1}{2\n\\sum_{n=1}^{N}\\left\\{ \\left[ P\\left( \\underline{p}\\right) +h\\left(\n\\underline{y},\\underline{x}\\right) \\right] ^{2}+\\left( \\frac{2\\pi }{T\n\\right) \\left[ Q\\left( \\underline{q}\\right) \\right] ^{2}\\right\\} ~.\n\\label{Htilde}\n\\end{equation\nIt can indeed be shown \\cite{CL2007,calogero} that it entails an \\textit\nisochronous} evolution (with period $T$) of the center of mass coordinate $Q\n, of the total momentum $P$, and---most importantly---of all the particle\ncoordinates, whose evolution then read\n\\begin{equation}\n\\underline{\\tilde{x}}\\left( t\\right) =\\underline{x}\\left( \\tau \\left(\nt\\right) \\right) ~,~~~\\underline{\\tilde{y}}\\left( t\\right) =\\underline{y\n\\left( \\tau \\left( t\\right) \\right) ~, \\label{xytildet}\n\\end{equation\nwhere we indicate as $\\underline{\\tilde{x}}\\left( t\\right) ,$ $\\underline\n\\tilde{y}}\\left( t\\right) $ the time evolution yielded by the tilded\nHamiltonian $\\tilde{H}\\left( \\underline{p},\\underline{q};T\\right) $ and as \n\\underline{x}\\left( t\\right) ,$ $\\underline{y}\\left( t\\right) $ the time\nevolution yielded by the original Hamiltonian $H\\left( \\underline{p}\n\\underline{q}\\right) ,$ an\n\\begin{equation}\n\\tau \\left( t\\right) =A~\\sin \\left( \\frac{2\\pi t}{T}+\\Phi \\right)\n\\label{taut}\n\\end{equation\nwhere $A$ and $\\Phi $ are \\textit{constant} parameters given by simple\nexpressions in terms of the initial values, $Q\\left( 0\\right) $ and $P\\left(\n0\\right) $, of the position of the center of mass of the system and of its\ntotal momentum, and of the Hamiltonian $\\tilde{H}\\left( \\underline{p}\n\\underline{q};T\\right) $ (which is of course a constant of motion for this\ntime evolution). Note that, for $\\left\\vert t\\right\\vert <\\tilde{T}$ in the\ntime coordinate $t\\,$); but it is \\textit{degenerate} at an infinite,\ndiscrete sequence of times $t_{n}=t_{0}\\pm nT\/2,$ $n=0,1,2,...$ . We\ninterpreted these metrics as corresponding to \\textit{isochronous\ncosmologies }\\cite{isochronous cosmologies}.\n\nLet us also recall \\cite{isochronous cosmologies} that this result is not\nrestricted to homogeneous, isotropic and spatially flat metrics: it can be\neasily extended to any synchronous metric, therefore it is quite general\nsince most metrics can be written in synchronous form by a diffeomorphic\nchange of coordinates.\n\nDue to the diffeomorphic correspondence between $g_{\\mu \\nu }$ and $\\tilde{g\n_{\\mu \\nu }$ \\textit{locally} in time---for time intervals of order $\\tilde{\n}$---these two metrics give the same physics locally in time, and therefore\nthere is no way to distinguish them using observations local in time; which\nis essentially the same finding valid in the context of the Hamiltonian\nsystems considered in \\cite{CL2007,calogero} and tersely recalled above. In\nparticular, the metric $\\tilde{g}_{\\mu \\nu }$---while being cyclic on time\nscales larger than $\\tilde{T}$---may yield over the time interval $\\tilde{T}$\njust the standard sequence of domination firstly by radiation, then by\nmatter and dark energy, as well as the cosmological perturbations consistent\nwith all observational tests \\cit\n{cmb,leansing,bao,lss,supernovae,ref0,ref1,ref2,ref3,ref4,bicep2}, which\ncharacterize the metric $g_{\\mu \\nu }$ of the standard $\\Lambda $-CDM\ncosmological model (see for instance \\cite{mukhanov} for a review).\nFurthermore, it has been shown that such isochronous metrics $\\tilde{g}_{\\mu\n\\nu }$ can be manufactured to be geodesically complete \\cite{isochronous\ncosmologies} and therefore singularity-free\\footnote\nA spacetime is singularity-free if it is geodesically complete, i.e. if its\ngeodesics can be always past- and future-extended \\cite{poisson,MTW}.}, so\nthat the geodesic motion as well as all physical quantities described by\nscalar invariants are always well defined \\cite{isochronous cosmologies} and\nthe Big Bang singularity may be avoided---even when some of the\nphenomenological observations can nevertheless be interpreted as remnants of\na past Big Bang, which however may never be actually attained by the metric \n\\tilde{g}_{\\mu \\nu }$, neither in the past nor in the future. This implies\nthat these isochronous metrics $\\tilde{g}_{\\mu \\nu }$ may describe a\nsingularity-free universe which---while featuring a time evolution which\nreproduces identically (up to diffeomorphic time reparameterization; \\textit\nlocally} in time, except at a discrete set of instants $t_{n}$) the standard\ncosmological model characterized by the metric $g_{\\mu \\nu }$---features an\nexpansion which stops at some instant, to be followed by a period of\ncontraction, until this phase of evolution stops and is again followed by an\nexpanding phase, this pattern being repeated \\textit{ad infinitum}. Let us\nre-emphasize that the universe characterized by the metric $\\tilde{g}_{\\mu\n\\nu }$ may thereby avoid to experience the Big Bang singularity, even if the\nuniverse which it mimics locally in time, characterized by the metric \ng_{\\mu \\nu },$ does encounter that singularity. Let us also note that this\nclass of metrics realize \\textit{de facto} the reversal of time's arrow, as\ndiscussed for instance in \\cite{sakharov}.\n\nBut in order to do so, these isochronous metrics must be \\textit{degenerate}\nat a discrete set of instants $t_{n}$ when the time reversals occur, say\nfrom expansion to contraction and viceversa; a phenomenology whose inclusion\nin general relativity might be considered problematic, because at these\ninstants $t_{n}$---when the metric is \\textit{degenerate}---the \"equivalence\nprinciple\" corresponding to the requirement that the metric tensor have a\nMinkowskian signature is violated. On the other hand a \\textit{physically\nunobservable} violation of a \"principle\" can be hardly considered \\textit\nphysically relevant}.\n\nPurpose and scope of this paper is to better clarify the meaning and the\ninterpretation of the \\textit{isochronous spacetimes} introduced in \\cit\n{isochronous cosmologies}, and to discuss some additional technical aspects.\nAs stated above, these isochronous metrics have the property to be \\textit\ndegenerate} at an infinite set of times $t_{n}$, i.e. over an infinite\nnumber of $3$-dimensional hypersurfaces, where generally the scale factor\nreaches its maximum and minimum values and the spacetime changes from an\nexpanding to a contracting phase and vice versa \\cite{isochronous\ncosmologies}.\n\nThe inclusion of degenerate metrics in general relativity is not a trivial\nmatter; indeed, it is forbidden if it were required---as an absolute (as it\nwere, \"metaphysical\") rule---that general relativity be consistent with the\n\"equivalence principle\", i. e. that spacetime \\textit{always} be a\nPseudo-Riemannian manifold \\textit{with Minkowskian signature}. This would\nindeed imply that the solutions of Einstein's general relativity, to be\nacceptable, must be \\textit{nondegenerate} metric tensors, since wherever in\nspacetime a metric is \\textit{degenerate} it is not possible to define its\ninverse, hence the Christoffel symbols as well as the Riemann, Ricci and\nEinstein tensors are not defined there. Hence---so the argument of some\ncritics goes---our isochronous metrics, which are not Minkowskian (locally,\non the hypersurfaces $t=t_{n}$) should be discarded in general relativity\nbecause they violate the equivalence principle at those degeneracy surfaces.\nOur rejoinder is that the metrics we introduced \\cite{isochronous\ncosmologies} are solutions of a version of general relativity whose\ndynamics, while still governed by Einstein's equations\\footnote\nEinstein's equations by itself do not determine the signature of spacetime,\nsee the discussion in \\cite{teitelboim,ellis1}.}, does allow the violation\nof the equivalence principle at a discrete set of hypersurfaces. It can be\nargued that such theories are therefore \\textit{different} from standard\ngeneral relativity, if validity of the equivalence principle---enforced as\nan absolute, universal rule---is considered an essential element of that\ntheory; although the difference is in fact \\textit{physically unobservable}.\nThis situation is indeed quite analogous with the findings described above,\nvalid in the nonrelativistic context of Hamiltonian many-body problems\n(classical or quantal), where the solutions of a physical theory described\nby a Hamiltonian $H$ are arbitrarily well approximated or even identically\nreproduced over an arbitrary time interval by periodic solutions of a\n\\textit{different} Hamiltonian $\\tilde{H}$, i.e. by a \\textit{different}\nphysical theory. The main difference is that in the general relativistic\ncontext in\\textbf{\\ }the two \\textit{different} physical theories the\ndynamic is given by the\\textit{\\ same} Einstein's equations, only the class\nof acceptable solutions is \\textit{different}.\n\nThe consideration in the context of general relativity of degenerate metrics\nis in any case not a novelty. For instance they were already introduced \\cit\n{ellis1,ellis2}, and subsequently investigated in a series of papers \\cit\n{elliscarfora,elliscomment,ellis3,ellis4,ellis5,ellis6,ellis8,ellis9,ellis10,ellis11,ellis12,ellis13,ellis14,ellis15,ellis16,ellis17,ellis18,ellis19}\n, in order to investigate signature-changing spacetimes. This class of\nsignature-changing metrics corresponds to a classical realization of the\nchange of signature in quantum cosmology conjectured by Hartle and Hawking\n\\cite{hartlehawking3} (see also \\cit\n{hartlehawking1,hartlehawking2,hartlehawking4}), which has philosophical\nimplications on the origin of the universe \\cite{hawkinghistory}.\n\nBelow we will also consider a different realization of isochronous\ncosmologies via non-degenerate metrics featuring a jump in their first\nderivatives at the inversion times $t_{n}$, which then implies a\ndistributional contribution in the stress-energy tensor at $t_{n}$. These\nmetrics are diffeomorphic to the continuous and degenerate isochronous\nmetrics everywhere except on the inversion hypersurfaces $t=t_{n}$, hence\nthey describe the same physics except at the infinite set of discrete times \nt_{n};$ therefore they may also be manufactured so as to agree with all\ncosmological observations (\\textit{locally} in time). Since these two\nrealizations of isochronous cosmologies are \\textit{locally} but \\textit{not\nglobally} diffeomorphic, they correspond to different spacetimes and may be\nconsidered to emerge from \\textit{different} theories: the nondegenerate\nones are generalized (in the sense of distributions) solutions of\nEinsteinian general relativity (including universal validity of the\nequivalence principle, but allowing jumps---whose physical significance and\njustification is moot---at the discrete times $t_{n}$); the degenerate ones\nfeature no mathematical or physical pathologies but fail to satisfy the\n\"equivalence principle\" at the discrete times $t_{n}$. They are essentially\nindistinguishable---among themselves and from non-isochronous\ncosmologies---by feasible experiments (unless one considers feasible an\nexperiment lasting an arbitrarily long time...).\n\n\\section{Isochronous Cosmologies}\n\nThe procedure used to construct, for an autonomous dynamical system $D$,\nanother autonomous dynamical system $\\tilde{D}$ whose solutions \\textit\napproximate} arbitrarily closely, or even reproduce exactly, those of the\nsystem $D$ over an \\textit{arbitrary} time interval $\\tilde{T}$ but are\n\\textit{isochronous} with an \\textit{arbitrary} period $T>\\tilde{T}$, is\nbased on the introduction of an auxiliary variable $\\tau $ in place of the\nphysical time variable $t$ \\cite{CL2007,calogero}. This procedure formally\nentails a change in the time dependence of any physical quantity $f(t)$, so\nthat $f(t)$ gets replaced by $f(\\tau \\left( t\\right) )$ with $\\tau \\left(\nt\\right) $ a periodic function of $t$ (with period $T$: $\\tau \\left( t\\pm\nT\\right) =\\tau \\left( t\\right) $), hence the new dynamics entails that all\nphysical quantities evolve periodically with period $T$. This change, in the\ncase of a nonrelativistic dynamical system, implies a change in the\ndynamical equations: for instance, in the case of the (quite general)\nmany-body problem described above, the theory is characterized by a new\n(autonomous) Hamiltonian $\\tilde{H}$---different from the original\n(autonomous) Hamiltonian $H$---which causes any physical variable originally\nevolving as $f\\left( t\\right) $ under the dynamics yielded by $H$, to evolve\ninstead as $f(\\tau (t))$ under the dynamics yielded by $\\tilde{H}$. This\nchange of the physical laws determining the time evolution of the system\nproduces the \\textit{isochronous} evolution \\cite{calogero,CL2007}. But let\nus stress---as we already did in \\cite{isochronous cosmologies}---that any\nattempt to attribute to the new variable $\\tau $ the significance of \"time\"\nwould be improper and confusing: the variable playing the role of \"time\" is\nthe same, $t$, for both the original dynamical system $D$ and the modified\ndynamical system $\\tilde{D}$; in particular, for that characterized by the\nstandard many-body Hamiltonian $H,$ see (\\ref{H}), as well as for that\ncharacterized by the modified many-body Hamiltonian $\\tilde{H}$, see (\\re\n{Htilde}).\n\nThe dynamical systems $D$ and $\\tilde{D}$ mentioned above feature the same\ntrajectories in phase space, but while the time evolution of the dynamical\nsystem $D$ corresponds to a \\textit{uniform} forward motion along those\ntrajectories, the time evolutions of the modified dynamical systems $\\tilde{\n}$---although produced by time-independent equations of motion---correspond\nto a \\textit{periodic} (with assigned period $T$), forward and backward,\ntime evolution along those same trajectories, exploring therefore only a\nportion of them; with the possibility to manufacture $\\tilde{D}$ in such a\nway that, for an \\textit{arbitrary} subinterval $\\tilde{T}0$ for any $t\n. This choice implies that all the scalar invariants as well as the pressure\nand energy density of the fluid remain finite at any $t$ and the spacetime\nis geodesically complete, thus singularity free.\n\n(vi) Geodesics have, in spacetime, a helical structure.\n\nThese solutions of course violate the \"equivalence principle\" at the\ninfinite set of discrete times $t_{n}.$ It is thus seen that, with the\nchoice (\\ref{bBperiodic}), the class of spacetime solutions of the equations\nof general relativity is enlarged by allowing unobservable violations of the\n\"equivalence principle\" at a discrete, numerable set of times $t_{n}$. This\nmore general class of spacetimes do not seem to entail any \\textit\nexperimentally observable} differences; although they might entail a quite\ndifferent structure of spacetime, for instance absence versus presence of\nBing Bang singularities.\n\nThe problem of finding a generalization of general relativity which includes\ndegenerate metrics has been studied in the past, motivated by the\nconsideration of a class of metrics, the so called signature-changing\nmetrics \\cit\n{ellis1,ellis2,elliscarfora,elliscomment,ellis3,ellis4,ellis5,ellis6,ellis8,ellis9,ellis10,ellis11,ellis12,ellis13,ellis14,ellis15,ellis16,ellis17,ellis18,ellis19}\n, which give a classical realization of the change of signature in quantum\ncosmology conjectured by Hartle and Hawking \\cit\n{hartlehawking3,hartlehawking1,hartlehawking2,hartlehawking4,hawkinghistory\n. The classical change of signature for a homogeneous, isotropic and\nspatially flat universe is realized by a metric tensor defined by the\nfollowing line elemen\n\\begin{equation}\nds^{2}=N(t)~dt^{2}-R(t)^{2}~d\\vec{x}^{2}~,\n\\label{changeofsignaturedegenerate}\n\\end{equation\nwhere $N(t)$ is a continuous function changing sign at some time $t_{0}$,\nfor instance $N(t)$ is positive for $t>t_{0}$, negative for $t\\tau _{0}$ and \nf(\\tau )=-1$ for $\\tau <\\tau _{0}$. It has been shown \\cite{ellis1,ellis2}\nthat in this case it is possible to introduce a smooth (generalized)\northonormal reference frame which allows a variational derivation of\nEinstein's equations and an extension of the Darmois formalism \\cite{ellis16}\nto discontinuous metrics, in such a way that, also in the case of a\ndiscontinuous metric tensor, the discontinuity of the extrinsic curvature on\na hypersurface is related to a distributional stress energy tensor \\cit\n{ellis15}. This is achieved by adding a surface term to the Einstein-Hilbert\naction, in such a way that the gravitational action become\n\\begin{equation}\nS_{g}=\\int_{M\/\\Sigma }\\sqrt{-g}Rd^{4}x+\\oint_{\\Sigma }\\sqrt{-g}Kd\\Sigma\n\\label{actionboundary}\n\\end{equation\nwhere $M$ is the manifold defining the spacetime, $\\Sigma $ is the boundary\nof $M$ where the metric tensor is discontinuous, $R$ is the Ricci scalar\ncurvature and $K$ the extrinsic curvature of $\\Sigma $. The addition of such\na boundary term is always necessary when one considers manifolds with\nboundaries \\cite{HawkingHorowitz}. Incidentally we note that this action\ncannot be used in the case of degenerate metrics, because in this case the\ndegeneracy surface $\\Sigma $ has no unitary normal vector hence the\nextrinsic curvature $K$ does not exist.\n\nThe two realizations of the classical change of signature, the continuous\nand degenerate one, see (\\ref{changeofsignaturedegenerate}), and the\ndiscontinuous one, see (\\ref{changeofsignaturediscontinuous}), are \\textit\nlocally but not globally} diffeomorphic, since they are related by a change\nof the time variable $dt=d\\tau \/\\sqrt{|N(t)|}$ which is not defined at the\ntime of signature change $t_{0}$ when $N(t)=0$. Hence they represent two\ndifferent spacetimes. However, except for the instants when $N(t)$ vanishes,\nthey are \\textit{locally } diffeomorphic and thus they describe---excepts at\nthose instants---the same physics .\n\nAs in the case of signature-changing metrics, it is also possible to\nconsider a realization of \\textit{isochronous} cosmologies different from\nthat we introduced in \\cite{isochronous cosmologies} and reviewed above (see\n(\\ref{newmetric}) and the subsequent treatment), via \\textit{nondegenerate}\nmetrics featuring a finite jump of their first derivatives at an infinite,\ndiscrete set of equispaced times. Such a realization is given by the\nfollowing metric\n\\begin{equation}\nds^{2}=d\\eta ^{2}-\\tilde{a}(\\eta )^{2}d\\vec{x}^{2}\n\\label{newmetricnondegenerate}\n\\end{equation\nwith\n\\begin{equation}\n\\alpha (\\tau (\\eta ))\\equiv \\tilde{a}(\\eta )\n\\label{newscalarfactornondegenerate}\n\\end{equation\nand\n\\begin{equation}\nd\\tau =C(\\eta )~d\\eta , \\label{newCndegenerate}\n\\end{equation\nwith a periodic function $C(\\eta )$ of period $T$ given for instance by \nC(\\eta )=1$ for $nT<\\eta <(2n+1)T\/2$ and $C(\\eta )=-1$ for $(2n+1)T\/2<\\eta\n<(n+1)T$ and $n$ integer, hence such that\n\\begin{subequations}\n\\begin{equation}\n\\tau \\left( \\eta \\right) \\equiv \\int_{0}^{\\eta }C\\left( \\eta ^{\\prime\n}\\right) d\\eta ^{\\prime }=\\eta -nT~~~\\text{for}~~~nT