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Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise 具有广义时空高斯噪声的抛物型Anderson模型的空间渐近性
IF 2.3 1区 数学
Annals of Probability Pub Date : 2016-03-01 DOI: 10.1214/15-AOP1006
Xia Chen
{"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}
引用次数: 44
Compensated fragmentation processes and limits of dilated fragmentations 补偿破碎过程和扩展破碎的极限
IF 2.3 1区 数学
Annals of Probability Pub Date : 2016-03-01 DOI: 10.1214/14-AOP1000
J. Bertoin
{"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}
引用次数: 20
Central limit theorem for linear groups 线性群的中心极限定理
IF 2.3 1区 数学
Annals of Probability Pub Date : 2016-03-01 DOI: 10.1214/15-AOP1002
Y. Benoist, Jean-François Quint
{"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}
引用次数: 87
On the perimeter of excursion sets of shot noise random fields 散粒噪声随机场偏移集的周长
IF 2.3 1区 数学
Annals of Probability Pub Date : 2016-01-01 DOI: 10.1214/14-AOP980
H. Biermé, A. Desolneux
{"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}
引用次数: 24
Interlacements and the wired uniform spanning forest 交错和有线均匀跨越森林
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-12-28 DOI: 10.1214/17-AOP1203
Tom Hutchcroft
{"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}
引用次数: 29
Size biased couplings and the spectral gap for random regular graphs 随机正则图的尺寸偏置耦合和谱隙
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-10-20 DOI: 10.1214/17-AOP1180
Nicholas A. Cook, L. Goldstein, Tobias Johnson
{"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}
引用次数: 51
The chaotic representation property of compensated-covariation stable families of martingales 补偿协变鞅稳定族的混沌表示性质
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-09-29 DOI: 10.1214/15-AOP1066
P. D. Tella, H. Engelbert
{"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}
引用次数: 9
Polarity of points for Gaussian random fields 高斯随机场点的极性
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-05-20 DOI: 10.1214/17-AOP1176
R. Dalang, C. Mueller, Yimin Xiao
{"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}
引用次数: 26
Wired cycle-breaking dynamics for uniform spanning forests 均匀跨越森林的有线循环打破动力学
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-04-15 DOI: 10.1214/15-AOP1063
Tom Hutchcroft
{"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}
引用次数: 15
Intermittency and multifractality: A case study via parabolic stochastic PDEs 间歇性和多重分形:一个基于抛物型随机偏微分方程的案例研究
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-03-20 DOI: 10.1214/16-AOP1147
D. Khoshnevisan, Kunwoo Kim, Yimin Xiao
{"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}
引用次数: 56
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