{"title":"A remarkable sigma-finite measure unifying supremum penalisations for a stable Lévy process","authors":"Yuko Yano","doi":"10.1214/12-AIHP497","DOIUrl":"https://doi.org/10.1214/12-AIHP497","url":null,"abstract":"","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"135 1","pages":"1014-1032"},"PeriodicalIF":1.5,"publicationDate":"2013-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88851822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 study the size of the giant component CG in the random geometric graph G = G(n, rn, f) of n nodes independently distributed each according to a certain density f(.) in [0, 1]2 satisfying infx∈[0,1]2 f(x) > 0. If c1 n ≤ r 2 n ≤ c2 logn n for some positive constants c1, c2 and nr 2 n −→ ∞, we show that the giant component of G contains at least n − o(n) nodes with probability at least 1 − o(1) as n → ∞. We also obtain estimates on the diameter and number of the non-giant components of G.
{"title":"Size of the giant component in a random geometric graph","authors":"Ghurumuruhan Ganesan","doi":"10.1214/12-AIHP498","DOIUrl":"https://doi.org/10.1214/12-AIHP498","url":null,"abstract":"In this paper, we study the size of the giant component CG in the random geometric graph G = G(n, rn, f) of n nodes independently distributed each according to a certain density f(.) in [0, 1]2 satisfying infx∈[0,1]2 f(x) > 0. If c1 n ≤ r 2 n ≤ c2 logn n for some positive constants c1, c2 and nr 2 n −→ ∞, we show that the giant component of G contains at least n − o(n) nodes with probability at least 1 − o(1) as n → ∞. We also obtain estimates on the diameter and number of the non-giant components of G.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"48 1","pages":"1130-1140"},"PeriodicalIF":1.5,"publicationDate":"2013-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80190923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
First, noncompact Cantor sets along with their defining trees are introduced as a natural generalization of p-adic numbers. Secondly we construct a class of jump processes on a noncompact Cantor set from given pairs of eigenvalues and measures. At the same time, we have concrete expressions of the associated jump kernels and transition densities. Then we construct intrinsic metrics on noncompact Cantor set to obtain estimates of transition densities and jump kernels under some regularity conditions on eigenvalues and measures . Finally transient random walks on the defining tree are shown to induce a subclass of jump processes discussed in the second part.
{"title":"Transitions on a noncompact Cantor set and random walks on its defining tree","authors":"Jun Kigami","doi":"10.1214/12-AIHP496","DOIUrl":"https://doi.org/10.1214/12-AIHP496","url":null,"abstract":"First, noncompact Cantor sets along with their defining trees are introduced as a natural generalization of p-adic numbers. Secondly we construct a class of jump processes on a noncompact Cantor set from given pairs of eigenvalues and measures. At the same time, we have concrete expressions of the associated jump kernels and transition densities. Then we construct intrinsic metrics on noncompact Cantor set to obtain estimates of transition densities and jump kernels under some regularity conditions on eigenvalues and measures . Finally transient random walks on the defining tree are shown to induce a subclass of jump processes discussed in the second part.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"104 1","pages":"1090-1129"},"PeriodicalIF":1.5,"publicationDate":"2013-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77381135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we prove the pathwise uniqueness for stochastic differential equations in Rd with time-dependent Sobolev drifts, and driven by symmetric α-stable processes provided that α ∈ (1,2) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α ∈ ( 2d d+1 ,2). Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales. Résumé. Dans cet article nous prouvons l’existence et l’unicité d’équations différentielles stochastiques dans Rd avec terme de dérive dépendant du temps dans un espace de Sobolev et dirigées par un processus de Lévy α-stable symétrique avec α ∈ (1,2) et de mesure spectrale non-dégénérée. En particulier, le terme de dérive peut avoir des discontinuités de saut quand α ∈ ( 2d d+1 ,2). Notre preuve est basée sur des estimations de type Krylov pour des semimartingales purement discontinues. MSC: 60H10
{"title":"Stochastic differential equations with Sobolev drifts and driven by $alpha$-stable processes","authors":"Xicheng Zhang","doi":"10.1214/12-AIHP476","DOIUrl":"https://doi.org/10.1214/12-AIHP476","url":null,"abstract":"In this article we prove the pathwise uniqueness for stochastic differential equations in Rd with time-dependent Sobolev drifts, and driven by symmetric α-stable processes provided that α ∈ (1,2) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α ∈ ( 2d d+1 ,2). Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales. Résumé. Dans cet article nous prouvons l’existence et l’unicité d’équations différentielles stochastiques dans Rd avec terme de dérive dépendant du temps dans un espace de Sobolev et dirigées par un processus de Lévy α-stable symétrique avec α ∈ (1,2) et de mesure spectrale non-dégénérée. En particulier, le terme de dérive peut avoir des discontinuités de saut quand α ∈ ( 2d d+1 ,2). Notre preuve est basée sur des estimations de type Krylov pour des semimartingales purement discontinues. MSC: 60H10","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"27 1","pages":"1057-1079"},"PeriodicalIF":1.5,"publicationDate":"2013-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72830595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-Holder functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichmuller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicom-pactness in Holder or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-Holder function space is particularly useful for investigating return time statistics. In particular we establish the shrinking target property for nested balls in the setting of Teichmuller flow. Our point of view, approach and terminology derive from the work of M. Pollicott augmented by that of M. Viana.
{"title":"Recurrence statistics for the space of interval exchange maps and the Teichmüller flow on the space of translation surfaces","authors":"R. Aimino, M. Nicol, M. Todd","doi":"10.1214/16-AIHP758","DOIUrl":"https://doi.org/10.1214/16-AIHP758","url":null,"abstract":"In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-Holder functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichmuller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicom-pactness in Holder or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-Holder function space is particularly useful for investigating return time statistics. In particular we establish the shrinking target property for nested balls in the setting of Teichmuller flow. Our point of view, approach and terminology derive from the work of M. Pollicott augmented by that of M. Viana.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"17 1","pages":"1371-1401"},"PeriodicalIF":1.5,"publicationDate":"2013-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85144852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The second author was supported by a Marie Curie IEF Fellowship, the third author acknowledges the support by Nizhny Novgorod University through the grant RNF 14-41-00044, and the fourth author was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1. This research has been supported by EU Marie-Curie IRSES Brazilian–European Partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS) and EU Marie-Sklodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-2014-ITN 643073 CRITICS).
{"title":"The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise","authors":"M. Callaway, T. S. Doan, J. Lamb, M. Rasmussen","doi":"10.1214/16-AIHP763","DOIUrl":"https://doi.org/10.1214/16-AIHP763","url":null,"abstract":"The second author was supported by a Marie Curie IEF Fellowship, the third author acknowledges the support by Nizhny Novgorod University \u0000through the grant RNF 14-41-00044, and the fourth author was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1. This research \u0000has been supported by EU Marie-Curie IRSES Brazilian–European Partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 \u0000BREUDS) and EU Marie-Sklodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-2014-ITN 643073 CRITICS).","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"67 1","pages":"1548-1574"},"PeriodicalIF":1.5,"publicationDate":"2013-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74108343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We continue the study initiated by Jean Bertoin in 2012 of a random dynamics on the edges of a uniform Cayley tree with $n$ vertices in which, successively, each edge is either set on fire with some fixed probability $p_n$ or fireproof with probability $1-p_n$. An edge which is set on fire burns and sets on fire its flammable neighbors, the fire then propagates in the tree, only stopped by fireproof edges. We study the distribution of the proportion of burnt and fireproof vertices and the sizes of the burnt or fireproof connected components as $n to infty$ regarding the asymptotic behavior of $p_n$.
我们继续Jean Bertoin在2012年对具有$n$顶点的均匀Cayley树的边的随机动力学的研究，其中每条边依次以固定概率$p_n$着火或以概率$1-p_n$防火。被点燃的边缘燃烧并点燃其易燃的邻居，然后火焰在树木中传播，只有防火边缘才能阻止。根据$p_n$的渐近性，我们研究了燃烧和防火顶点的比例分布以及燃烧或防火连接分量的大小$n to infty$。
{"title":"On the sizes of burnt and fireproof components for fires on a large Cayley tree","authors":"Cyril Marzouk","doi":"10.1214/14-AIHP640","DOIUrl":"https://doi.org/10.1214/14-AIHP640","url":null,"abstract":"We continue the study initiated by Jean Bertoin in 2012 of a random dynamics on the edges of a uniform Cayley tree with $n$ vertices in which, successively, each edge is either set on fire with some fixed probability $p_n$ or fireproof with probability $1-p_n$. An edge which is set on fire burns and sets on fire its flammable neighbors, the fire then propagates in the tree, only stopped by fireproof edges. We study the distribution of the proportion of burnt and fireproof vertices and the sizes of the burnt or fireproof connected components as $n to infty$ regarding the asymptotic behavior of $p_n$.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"4 1","pages":"355-375"},"PeriodicalIF":1.5,"publicationDate":"2013-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80700063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of q-TASEP at time t are of order t^{1/3} and asymptotically distributed as the GUE Tracy-Widom distribution, which confirms the KPZ scaling theory conjecture.
{"title":"Tracy–Widom asymptotics for $q$-TASEP","authors":"P. Ferrari, B. Vető","doi":"10.1214/14-AIHP614","DOIUrl":"https://doi.org/10.1214/14-AIHP614","url":null,"abstract":"We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of q-TASEP at time t are of order t^{1/3} and asymptotically distributed as the GUE Tracy-Widom distribution, which confirms the KPZ scaling theory conjecture.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"32 1","pages":"1465-1485"},"PeriodicalIF":1.5,"publicationDate":"2013-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74441383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider weak invariance principles (functional limit theorems) in the domain of a stable law. A general result is obtained on lifting such limit laws from an induced dynamical system to the original system. An important class of examples covered by our result are Pomeau-Manneville intermittency maps, where convergence for the induced system is in the standard Skorohod J1 topology. For the full system, convergence in the J1 topology fails, but we prove convergence in theM1 topology.
{"title":"Weak Convergence to Stable Lévy Processes for Nonuniformly Hyperbolic Dynamical Systems","authors":"I. Melbourne, Roland Zweimuller","doi":"10.1214/13-AIHP586","DOIUrl":"https://doi.org/10.1214/13-AIHP586","url":null,"abstract":"We consider weak invariance principles (functional limit theorems) in the domain of a stable law. A general result is obtained on lifting such limit laws from an induced dynamical system to the original system. An important class of examples covered by our result are Pomeau-Manneville intermittency maps, where convergence for the induced system is in the standard Skorohod J1 topology. For the full system, convergence in the J1 topology fails, but we prove convergence in theM1 topology.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"12 1","pages":"545-556"},"PeriodicalIF":1.5,"publicationDate":"2013-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78783304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider an elliptic Kolmogorov equationu − Ku = f in a convex subset C of a separable Hilbert space X. The Kolmogorov operator K is a realization of u 7→ 1 Tr (D 2 u(x)) + hAx − DU(x),Du(x)i, A is a self-adjoint operator in X and U : X 7→R ∪ {+∞} is a convex function. We prove that for � > 0 and f ∈ L 2 (C,�) the weak solution u belongs to the Sobolev space W 2,2 (C,�), whereis the log-concave measure associated to the system. Moreover we prove maximal estimates on the gradient of u, that allow to show that u satisfies the Neumann boundary condition in the sense of traces at the boundary of C. The general results are applied to Kolmogorov equations of reaction-diffusion and Cahn-Hilliard stochastic PDEs in convex sets of suitable Hilbert spaces.
{"title":"Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains","authors":"G. Prato, A. Lunardi","doi":"10.1214/14-AIHP611","DOIUrl":"https://doi.org/10.1214/14-AIHP611","url":null,"abstract":"We consider an elliptic Kolmogorov equationu − Ku = f in a convex subset C of a separable Hilbert space X. The Kolmogorov operator K is a realization of u 7→ 1 Tr (D 2 u(x)) + hAx − DU(x),Du(x)i, A is a self-adjoint operator in X and U : X 7→R ∪ {+∞} is a convex function. We prove that for � > 0 and f ∈ L 2 (C,�) the weak solution u belongs to the Sobolev space W 2,2 (C,�), whereis the log-concave measure associated to the system. Moreover we prove maximal estimates on the gradient of u, that allow to show that u satisfies the Neumann boundary condition in the sense of traces at the boundary of C. The general results are applied to Kolmogorov equations of reaction-diffusion and Cahn-Hilliard stochastic PDEs in convex sets of suitable Hilbert spaces.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"34 1","pages":"1102-1123"},"PeriodicalIF":1.5,"publicationDate":"2013-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91240034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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