Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225–2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483–533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation {∂u∂t(t,x)=12Δu(t,x)+V(t,x)u(t,x),u(0,x)=u0(x), where the homogeneous generalized Gaussian noise V(t,x) is, among other forms, white or fractional white in time and space. Associated with the Cole–Hopf solution to the KPZ equation, in particular, the precise asymptotic form limR→∞(logR)−2/3logmax|x|≤Ru(t,x)=342t3−−−√3a.s. is obtained for the parabolic Anderson model ∂tu=12∂2xxu+W˙u with the (1+1)-white noise W˙(t,x). In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.
{"title":"Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise","authors":"Xia Chen","doi":"10.1214/15-AOP1006","DOIUrl":"https://doi.org/10.1214/15-AOP1006","url":null,"abstract":"Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225–2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483–533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation \u0000{∂u∂t(t,x)=12Δu(t,x)+V(t,x)u(t,x),u(0,x)=u0(x), \u0000where the homogeneous generalized Gaussian noise V(t,x) \u0000is, among other forms, white or fractional white in time and space. Associated with the Cole–Hopf solution to the KPZ equation, in particular, the precise asymptotic form \u0000limR→∞(logR)−2/3logmax|x|≤Ru(t,x)=342t3−−−√3a.s. \u0000is obtained for the parabolic Anderson model ∂tu=12∂2xxu+W˙u with the (1+1)-white noise W˙(t,x). In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"1535-1598"},"PeriodicalIF":2.3,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new class of fragmentation-type random processes is introduced, in which, roughly speaking, the accumulation of small dislocations which would instantaneously shatter the mass into dust, is compensated by an adequate dilation of the components. An important feature of these compensated fragmentations is that the dislocation measure $nu$ which governs their evolutions has only to fulfill the integral condition $int_{mathit{p}}$ (1-$mathit{p}_{1}$)$^{2}nu$(d$mathbf{p}$ < $infty$, where $mathbf{p}$ = ($mathit{p}_{1}$,…) denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $int_{mathit{p}}$ (1-$mathit{p}_{1}$)$^{2}nu$(d$mathbf{p}$ < $infty$ for $nu$ to be the dislocation measure of a homogeneous fragmentation. Our main results show that such compensated fragmentations naturally arise as limits of homogeneous dilated fragmentations, and bear close connections to spectrally negative Levy processes.
{"title":"Compensated fragmentation processes and limits of dilated fragmentations","authors":"J. Bertoin","doi":"10.1214/14-AOP1000","DOIUrl":"https://doi.org/10.1214/14-AOP1000","url":null,"abstract":"A new class of fragmentation-type random processes is introduced, in which, roughly speaking, the accumulation of small dislocations which would instantaneously shatter the mass into dust, is compensated by an adequate dilation of the components. An important feature of these compensated fragmentations is that the dislocation measure $nu$ which governs their evolutions has only to fulfill the integral condition $int_{mathit{p}}$ (1-$mathit{p}_{1}$)$^{2}nu$(d$mathbf{p}$ < $infty$, where $mathbf{p}$ = ($mathit{p}_{1}$,…) denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $int_{mathit{p}}$ (1-$mathit{p}_{1}$)$^{2}nu$(d$mathbf{p}$ < $infty$ for $nu$ to be the dislocation measure of a homogeneous fragmentation. Our main results show that such compensated fragmentations naturally arise as limits of homogeneous dilated fragmentations, and bear close connections to spectrally negative Levy processes.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"1254-1284"},"PeriodicalIF":2.3,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP1000","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66004903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a central limit theorem for random walks with finite variance on linear groups.
证明了线性群上有限方差随机漫步的一个中心极限定理。
{"title":"Central limit theorem for linear groups","authors":"Y. Benoist, Jean-François Quint","doi":"10.1214/15-AOP1002","DOIUrl":"https://doi.org/10.1214/15-AOP1002","url":null,"abstract":"We prove a central limit theorem for random walks with finite variance on linear groups.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"1308-1340"},"PeriodicalIF":2.3,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension n≥1. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.
{"title":"On the perimeter of excursion sets of shot noise random fields","authors":"H. Biermé, A. Desolneux","doi":"10.1214/14-AOP980","DOIUrl":"https://doi.org/10.1214/14-AOP980","url":null,"abstract":"In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension n≥1. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"521-543"},"PeriodicalIF":2.3,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP980","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66010247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the Aldous-Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman's random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of `excessive ends' in the WUSF is non-random in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm, while the third extends a recent result of the author. Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.
{"title":"Interlacements and the wired uniform spanning forest","authors":"Tom Hutchcroft","doi":"10.1214/17-AOP1203","DOIUrl":"https://doi.org/10.1214/17-AOP1203","url":null,"abstract":"We extend the Aldous-Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman's random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of `excessive ends' in the WUSF is non-random in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm, while the third extends a recent result of the author. \u0000Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"46 1","pages":"1170-1200"},"PeriodicalIF":2.3,"publicationDate":"2015-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1203","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66060935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let λλ be the second largest eigenvalue in absolute value of a uniform random dd-regular graph on nn vertices. It was famously conjectured by Alon and proved by Friedman that if dd is fixed independent of nn, then λ=2d−1−−−−√+o(1)λ=2d−1+o(1) with high probability. In the present work, we show that λ=O(d−−√)λ=O(d) continues to hold with high probability as long as d=O(n2/3)d=O(n2/3), making progress toward a conjecture of Vu that the bound holds for all 1≤d≤n/21≤d≤n/2. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at d=o(n1/2)d=o(n1/2). We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on dd-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.
{"title":"Size biased couplings and the spectral gap for random regular graphs","authors":"Nicholas A. Cook, L. Goldstein, Tobias Johnson","doi":"10.1214/17-AOP1180","DOIUrl":"https://doi.org/10.1214/17-AOP1180","url":null,"abstract":"Let λλ be the second largest eigenvalue in absolute value of a uniform random dd-regular graph on nn vertices. It was famously conjectured by Alon and proved by Friedman that if dd is fixed independent of nn, then λ=2d−1−−−−√+o(1)λ=2d−1+o(1) with high probability. In the present work, we show that λ=O(d−−√)λ=O(d) continues to hold with high probability as long as d=O(n2/3)d=O(n2/3), making progress toward a conjecture of Vu that the bound holds for all 1≤d≤n/21≤d≤n/2. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at d=o(n1/2)d=o(n1/2). We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on dd-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"46 1","pages":"72-125"},"PeriodicalIF":2.3,"publicationDate":"2015-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1180","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66060569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, we study the chaotic representation property for certain families XX of square integrable martingales on a finite time interval [0,T][0,T]. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family XX of square integrable martingales having deterministic mutual predictable covariation ⟨X,Y⟩⟨X,Y⟩ for all X,Y∈XX,Y∈X. The main result of the present paper is stated in Theorem 5.8 below: If XX is a compensated-covariation stable family of square integrable martingales such that ⟨X,Y⟩⟨X,Y⟩ is deterministic for all X,Y∈XX,Y∈X and, furthermore, the system of monomials generated by XX is total in L2(Ω,FXT,P)L2(Ω,FTX,P), then XX possesses the chaotic representation property with respect to the σσ-field FXTFTX. We shall apply this result to the case of Levy processes. Relative to the filtration FLFL generated by a Levy process LL, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Levy processes, several examples of concrete families XX of martingales including Teugels martingales.
{"title":"The chaotic representation property of compensated-covariation stable families of martingales","authors":"P. D. Tella, H. Engelbert","doi":"10.1214/15-AOP1066","DOIUrl":"https://doi.org/10.1214/15-AOP1066","url":null,"abstract":"In the present paper, we study the chaotic representation property for certain families XX of square integrable martingales on a finite time interval [0,T][0,T]. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family XX of square integrable martingales having deterministic mutual predictable covariation ⟨X,Y⟩⟨X,Y⟩ for all X,Y∈XX,Y∈X. The main result of the present paper is stated in Theorem 5.8 below: If XX is a compensated-covariation stable family of square integrable martingales such that ⟨X,Y⟩⟨X,Y⟩ is deterministic for all X,Y∈XX,Y∈X and, furthermore, the system of monomials generated by XX is total in L2(Ω,FXT,P)L2(Ω,FTX,P), then XX possesses the chaotic representation property with respect to the σσ-field FXTFTX. We shall apply this result to the case of Levy processes. Relative to the filtration FLFL generated by a Levy process LL, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Levy processes, several examples of concrete families XX of martingales including Teugels martingales.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"3965-4005"},"PeriodicalIF":2.3,"publicationDate":"2015-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1066","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66033099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space–time white noise, or colored noise in spatial dimensions k≥1k≥1. Our approach builds on a delicate covering argument developed by M. Talagrand [Ann. Probab. 23 (1995) 767–775; Probab. Theory Related Fields 112 (1998) 545–563] for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic PDEs.
{"title":"Polarity of points for Gaussian random fields","authors":"R. Dalang, C. Mueller, Yimin Xiao","doi":"10.1214/17-AOP1176","DOIUrl":"https://doi.org/10.1214/17-AOP1176","url":null,"abstract":"We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space–time white noise, or colored noise in spatial dimensions k≥1k≥1. Our approach builds on a delicate covering argument developed by M. Talagrand [Ann. Probab. 23 (1995) 767–775; Probab. Theory Related Fields 112 (1998) 545–563] for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic PDEs.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"4700-4751"},"PeriodicalIF":2.3,"publicationDate":"2015-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1176","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66060617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that every component of the wired uniform spanning forest (WUSFWUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSFWUSF is one-ended almost surely in every supercritical Galton–Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm [Ann. Probab. 29 (2001) 1–65]. Our proof introduces and exploits a family of Markov chains under which the oriented WUSFWUSF is stationary, which we call the wired cycle-breaking dynamics.
{"title":"Wired cycle-breaking dynamics for uniform spanning forests","authors":"Tom Hutchcroft","doi":"10.1214/15-AOP1063","DOIUrl":"https://doi.org/10.1214/15-AOP1063","url":null,"abstract":"We prove that every component of the wired uniform spanning forest (WUSFWUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSFWUSF is one-ended almost surely in every supercritical Galton–Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm [Ann. Probab. 29 (2001) 1–65]. \u0000 \u0000Our proof introduces and exploits a family of Markov chains under which the oriented WUSFWUSF is stationary, which we call the wired cycle-breaking dynamics.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"3879-3892"},"PeriodicalIF":2.3,"publicationDate":"2015-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66033231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let denote space-time white noise, and consider the following stochastic partial dierential equations: (i) _ u = 1 u 00 +u , started identically at one; and (ii) _ Z = 1 Z 00 + , started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in dierent universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on R+ R d with d > 2. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein{Uhlenbeck process on R are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S.J. Taylor [3, 4]. We expand on aspects of the Barlow{Taylor theory, as well.
设为时空白噪声，并考虑以下随机偏微分方程:(i) _ u = 1 u 00 +u，在1相等开始;和(ii) _ Z = 1z00 +，相同地从零开始。众所周知，(i)的解决方案是间歇性的，而(ii)的解决方案则不是。已知这两个方程属于不同的通用性类。我们证明了这两个体系的高峰在自然的大尺度意义上是多重分形的。在此基础上进一步推广了随机偏微分方程在R+ R+ d上的多重分形行为。G. Lawler问我们间歇性是否等同于多重分形。目前的工作对这个问题给出了否定的答案。作为我们的方法的副产品，我们还证明了布朗运动的峰形成一个大尺度的单分形，而R上的Ornstein{Uhlenbeck过程的峰是多重分形。自始至终，我们广泛运用了M.T. Barlow和S.J. Taylor的宏观分形理论[3,4]。我们也扩展了巴洛{泰勒理论的各个方面。
{"title":"Intermittency and multifractality: A case study via parabolic stochastic PDEs","authors":"D. Khoshnevisan, Kunwoo Kim, Yimin Xiao","doi":"10.1214/16-AOP1147","DOIUrl":"https://doi.org/10.1214/16-AOP1147","url":null,"abstract":"Let denote space-time white noise, and consider the following stochastic partial dierential equations: (i) _ u = 1 u 00 +u , started identically at one; and (ii) _ Z = 1 Z 00 + , started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in dierent universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on R+ R d with d > 2. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein{Uhlenbeck process on R are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S.J. Taylor [3, 4]. We expand on aspects of the Barlow{Taylor theory, as well.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"3697-3751"},"PeriodicalIF":2.3,"publicationDate":"2015-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1147","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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