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High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder 重尾无序$(1+1)$维定向聚合物的高温极限
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-03-03 DOI: 10.1214/15-AOP1067
P. Dey, Nikos Zygouras
{"title":"High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder","authors":"P. Dey, Nikos Zygouras","doi":"10.1214/15-AOP1067","DOIUrl":"https://doi.org/10.1214/15-AOP1067","url":null,"abstract":"The directed polymer model at intermediate disorder regime was introduced by Alberts–Khanin–Quastel [Ann. Probab. 42 (2014) 1212–1256]. It was proved that at inverse temperature βn−γβn−γ with γ=1/4γ=1/4 the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of γγ.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"4006-4048"},"PeriodicalIF":2.3,"publicationDate":"2015-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66033162","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: 21
Path-dependent equations and viscosity solutions in infinite dimension 无限维的路径相关方程和粘性解
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-02-19 DOI: 10.1214/17-AOP1181
Andrea Cosso, S. Federico, Fausto Gozzi, M. Rosestolato, N. Touzi
{"title":"Path-dependent equations and viscosity solutions in infinite dimension","authors":"Andrea Cosso, S. Federico, Fausto Gozzi, M. Rosestolato, N. Touzi","doi":"10.1214/17-AOP1181","DOIUrl":"https://doi.org/10.1214/17-AOP1181","url":null,"abstract":"Path Dependent PDE's (PPDE's) are natural objects to study when one deals with non Markovian models. Recently, after the introduction (see [12]) of the so-called pathwise (or functional or Dupire) calculus, various papers have been devoted to study the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [18]) and viscosity solutions (see e.g. [13]), in the case of finite dimensional underlying space. In this paper, motivated by the study of models driven by path dependent stochastic PDE's, we give a first well-posedness result for viscosity solutions of PPDE's when the underlying space is an infinite dimensional Hilbert space. The proof requires a substantial modification of the approach followed in the finite dimensional case. We also observe that, differently from the finite dimensional case, our well-posedness result, even in the Markovian case, apply to equations which cannot be treated, up to now, with the known theory of viscosity solutions.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"46 1","pages":"126-174"},"PeriodicalIF":2.3,"publicationDate":"2015-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1181","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66060651","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: 36
Climbing down Gaussian peaks 沿着高斯峰爬升
IF 2.3 1区 数学
Annals of Probability Pub Date : 2015-01-28 DOI: 10.1214/15-AOP1083
R. Adler, G. Samorodnitsky
{"title":"Climbing down Gaussian peaks","authors":"R. Adler, G. Samorodnitsky","doi":"10.1214/15-AOP1083","DOIUrl":"https://doi.org/10.1214/15-AOP1083","url":null,"abstract":"Research supported in part by US-Israel Binational \u0000Science Foundation, 2008262, by ARO \u0000grant W911NF-12-10385, NSF grant DMS-1005903 and URSAT, ERC Advanced Grant 320422","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"1160-1189"},"PeriodicalIF":2.3,"publicationDate":"2015-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1083","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66033129","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: 7
A lower bound for disconnection by simple random walk 简单随机漫步断连的下界
IF 2.3 1区 数学
Annals of Probability Pub Date : 2014-12-12 DOI: 10.1214/15-AOP1077
Xinyi Li
{"title":"A lower bound for disconnection by simple random walk","authors":"Xinyi Li","doi":"10.1214/15-AOP1077","DOIUrl":"https://doi.org/10.1214/15-AOP1077","url":null,"abstract":"We consider simple random walk on ZdZd, d≥3d≥3. Motivated by the work of A.-S. Sznitman and the author in [Probab. Theory Related Fields 161 (2015) 309–350] and [Electron. J. Probab. 19 (2014) 1–26], we investigate the asymptotic behavior of the probability that a large body gets disconnected from infinity by the set of points visited by a simple random walk. We derive asymptotic lower bounds that bring into play random interlacements. Although open at the moment, some of the lower bounds we obtain possibly match the asymptotic upper bounds recently obtained in [Disconnection, random walks, and random interlacements (2014)]. This potentially yields special significance to the tilted walks that we use in this work, and to the strategy that we employ to implement disconnection.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"879-931"},"PeriodicalIF":2.3,"publicationDate":"2014-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1077","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032936","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: 11
Central limit theorems and bootstrap in high dimensions 中心极限定理与高维自举
IF 2.3 1区 数学
Annals of Probability Pub Date : 2014-12-11 DOI: 10.1920/WP.CEM.2016.3916
V. Chernozhukov, D. Chetverikov, Kengo Kato
{"title":"Central limit theorems and bootstrap in high dimensions","authors":"V. Chernozhukov, D. Chetverikov, Kengo Kato","doi":"10.1920/WP.CEM.2016.3916","DOIUrl":"https://doi.org/10.1920/WP.CEM.2016.3916","url":null,"abstract":"In this paper, we derive central limit and bootstrap theorems for probabilities that centered high-dimensional vector sums hit rectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for the probabilities that a root-n rescaled sample average of Xi is in A, where X1,..., Xn are independent random vectors in Rp and A is a rectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn-> infinity and p>>n; in particular, p can be as large as O(e^(Cn^c)) for some constants c,C>0. The result holds uniformly over all rectangles, or more generally, sparsely convex sets, and does not require any restrictions on the correlation among components of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend nontrivially only on a small subset of their arguments, with rectangles being a special case.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"2309-2352"},"PeriodicalIF":2.3,"publicationDate":"2014-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68013037","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: 281
Large deviation estimates for exceedance times of perpetuity sequences and their dual processes 永续序列超越次数的大偏差估计及其对偶过程
IF 2.3 1区 数学
Annals of Probability Pub Date : 2014-11-27 DOI: 10.1214/15-AOP1059
D. Buraczewski, Jeffrey F. Collamore, E. Damek, J. Zienkiewicz
{"title":"Large deviation estimates for exceedance times of perpetuity sequences and their dual processes","authors":"D. Buraczewski, Jeffrey F. Collamore, E. Damek, J. Zienkiewicz","doi":"10.1214/15-AOP1059","DOIUrl":"https://doi.org/10.1214/15-AOP1059","url":null,"abstract":"In a variety of problems in pure and applied probability, it is relevant to study the large exceedance probabilities of the perpetuity sequence Yn:=B1+A1B2+⋯+(A1⋯An−1)BnYn:=B1+A1B2+⋯+(A1⋯An−1)Bn, where (Ai,Bi)⊂(0,∞)×R(Ai,Bi)⊂(0,∞)×R. Estimates for the stationary tail distribution of {Yn}{Yn} have been developed in the seminal papers of Kesten [Acta Math. 131 (1973) 207–248] and Goldie [Ann. Appl. Probab. 1 (1991) 126–166]. Specifically, it is well known that if M:=supnYnM:=supnYn, then P{M>u}∼CMu−ξP{M>u}∼CMu−ξ as u→∞u→∞. While much attention has been focused on extending such estimates to more general settings, little work has been devoted to understanding the path behavior of these processes. In this paper, we derive sharp asymptotic estimates for the normalized first passage time Tu:=(logu)−1inf{n:Yn>u}Tu:=(log⁡u)−1inf{n:Yn>u}. We begin by showing that, conditional on {Tu<∞}{Tu<∞}, Tu→ρTu→ρ as u→∞u→∞ for a certain positive constant ρρ. We then provide a conditional central limit theorem for {Tu}{Tu}, and study P{Tu∈G}P{Tu∈G} as u→∞u→∞ for sets G⊂[0,∞)G⊂[0,∞). If G⊂[0,ρ)G⊂[0,ρ), then we show that P{Tu∈G}uI(G)→C(G)P{Tu∈G}uI(G)→C(G) as u→∞u→∞ for a certain large deviation rate function II and constant C(G)C(G). On the other hand, if G⊂(ρ,∞)G⊂(ρ,∞), then we show that the tail behavior is actually quite complex and different asymptotic regimes are possible. We conclude by extending our results to the corresponding forward process, understood in the sense of Letac [In Random Matrices and Their Applications (Brunswick, Maine, 1984) (1986) 263–273 Amer. Math. Soc.], namely to the reflected process M∗n:=max{AnM∗n−1+Bn,0}Mn∗:=max{AnMn−1∗+Bn,0}, n∈Z+n∈Z+. Using Siegmund duality, we relate the first passage times of {Yn}{Yn} to the finite-time exceedance probabilities of {M∗n}{Mn∗}, yielding a new result concerning the convergence of {M∗n}{Mn∗} to its stationary distribution.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"3688-3739"},"PeriodicalIF":2.3,"publicationDate":"2014-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1059","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032696","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
Dissipation and high disorder 耗散和高度失序
IF 2.3 1区 数学
Annals of Probability Pub Date : 2014-11-24 DOI: 10.1214/15-AOP1040
Le Chen, M. Cranston, D. Khoshnevisan, Kunwoo Kim
{"title":"Dissipation and high disorder","authors":"Le Chen, M. Cranston, D. Khoshnevisan, Kunwoo Kim","doi":"10.1214/15-AOP1040","DOIUrl":"https://doi.org/10.1214/15-AOP1040","url":null,"abstract":"Given a fieldfB(x)gx2Zd of independent standard Brownian motions, indexed by Z d , the generator of a suitable Markov process on Z d ; G; and suciently nice function : [0;1)! [0;1); we consider the influence of the parameter on the behavior of the system, dut(x) = (Gut)(x) dt + (ut(x))dBt(x) [t > 0; x2 Z d ]; u0(x) = c0 0(x); We show that for any","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"82-99"},"PeriodicalIF":2.3,"publicationDate":"2014-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1040","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031932","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: 4
Tail estimates for Markovian rough paths 马尔可夫粗糙路径的尾部估计
IF 2.3 1区 数学
Annals of Probability Pub Date : 2014-11-19 DOI: 10.1214/16-AOP1117
T. Cass, M. Ogrodnik
{"title":"Tail estimates for Markovian rough paths","authors":"T. Cass, M. Ogrodnik","doi":"10.1214/16-AOP1117","DOIUrl":"https://doi.org/10.1214/16-AOP1117","url":null,"abstract":"We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms ([26]) and prove a better-than-exponential tail estimate for the accumulated local p-variation functional, which has been introduced and studied in [17]. We comment on the significance of these estimates to a range of currently-studied problems, including the recent results of Ni Hao [32], and Chevyrev and Lyons [18].","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"2477-2504"},"PeriodicalIF":2.3,"publicationDate":"2014-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1117","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047729","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: 11
BSE’s, BSDE’s and fixed-point problems BSE, BSDE和不动点问题
IF 2.3 1区 数学
Annals of Probability Pub Date : 2014-10-06 DOI: 10.1214/16-AOP1149
Patrick Cheridito, Kihun Nam
{"title":"BSE’s, BSDE’s and fixed-point problems","authors":"Patrick Cheridito, Kihun Nam","doi":"10.1214/16-AOP1149","DOIUrl":"https://doi.org/10.1214/16-AOP1149","url":null,"abstract":"In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed-point problem in a space of random vectors. This makes it possible to employ general fixed-point arguments to establish the existence of a solution. For instance, Banach’s contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed-point arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean–Vlasov-type equations and equations with coefficients of superlinear growth.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"3795-3828"},"PeriodicalIF":2.3,"publicationDate":"2014-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1149","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047820","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: 19
A quantitative Burton-Keane estimate under strong FKG condition 强FKG条件下的定量Burton-Keane估计
IF 2.3 1区 数学
Annals of Probability Pub Date : 2014-09-18 DOI: 10.1214/15-AOP1049
H. Duminil-Copin, D. Ioffe, Y. Velenik
{"title":"A quantitative Burton-Keane estimate under strong FKG condition","authors":"H. Duminil-Copin, D. Ioffe, Y. Velenik","doi":"10.1214/15-AOP1049","DOIUrl":"https://doi.org/10.1214/15-AOP1049","url":null,"abstract":"We consider translationally-invariant percolation models on ZdZd satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance nn (this corresponds to a finite size version of the celebrated Burton–Keane [Comm. Math. Phys. 121 (1989) 501–505] argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincare inequality proved in Chatterjee and Sen (2013). As a consequence, we obtain upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight q≥1q≥1.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"3335-3356"},"PeriodicalIF":2.3,"publicationDate":"2014-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032735","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: 2
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