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Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations 随机穿孔域中非线性德里赫特问题的均质化（关于穿孔大小的最小假设条件
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-09-17 DOI: 10.1007/s00440-024-01320-1
Lucia Scardia, Konstantinos Zemas, Caterina Ida Zeppieri
{"title":"Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations","authors":"Lucia Scardia, Konstantinos Zemas, Caterina Ida Zeppieri","doi":"10.1007/s00440-024-01320-1","DOIUrl":"https://doi.org/10.1007/s00440-024-01320-1","url":null,"abstract":"<p>In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of <span>(mathbb {R}^n)</span>. Under the assumption that the perforations are small balls whose centres and radii are generated by a <i>stationary short-range marked point process</i>, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite <span>((n-q))</span>-moment, where <span>(1&lt;q&lt;n)</span> is the growth-exponent of the associated energy functionals. This assumption on the one hand ensures that the expectation of the nonlinear <i>q</i>-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257708","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: 0
On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements 关于维纳香肠空位集和布朗交错的唯一性问题
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-09-14 DOI: 10.1007/s00440-024-01315-y
Yingxin Mu, Artem Sapozhnikov
{"title":"On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements","authors":"Yingxin Mu, Artem Sapozhnikov","doi":"10.1007/s00440-024-01315-y","DOIUrl":"https://doi.org/10.1007/s00440-024-01315-y","url":null,"abstract":"<p>We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in <span>({mathbb {R}}^d)</span> in the transient dimensions <span>(d ge 3)</span>. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least 2 connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257709","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: 0
Weighted sums and Berry-Esseen type estimates in free probability theory 自由概率论中的加权和与贝里-埃森类型估计
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-08-21 DOI: 10.1007/s00440-024-01294-0
Leonie Neufeld
{"title":"Weighted sums and Berry-Esseen type estimates in free probability theory","authors":"Leonie Neufeld","doi":"10.1007/s00440-024-01294-0","DOIUrl":"https://doi.org/10.1007/s00440-024-01294-0","url":null,"abstract":"<p>We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner’s semicircle law is of order <span>(n^{-frac{1}{2}})</span> with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order <span>(n^{-1})</span>, thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalize to the setting of sums of free non-identically distributed bounded self-adjoint random variables leading to a new rate of convergence in the free central limit theorem.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142223921","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: 0
Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules 具有对称各向同性阈值规则的二维引导渗流的尖锐蜕变
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-08-21 DOI: 10.1007/s00440-024-01310-3
Hugo Duminil-Copin, Ivailo Hartarsky
{"title":"Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules","authors":"Hugo Duminil-Copin, Ivailo Hartarsky","doi":"10.1007/s00440-024-01310-3","DOIUrl":"https://doi.org/10.1007/s00440-024-01310-3","url":null,"abstract":"<p>We study two-dimensional critical bootstrap percolation models. We establish that a class of these models including all isotropic threshold rules with a convex symmetric neighbourhood, undergoes a sharp metastability transition. This extends previous instances proved for several specific rules. The paper supersedes a draft by Alexander Holroyd and the first author from 2012. While it served a role in the subsequent development of bootstrap percolation universality, we have chosen to adopt a more contemporary viewpoint in its present form.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182572","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: 0
Subexponential lower bounds for f-ergodic Markov processes f-ergodic Markov 过程的亚指数下界
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-08-21 DOI: 10.1007/s00440-024-01306-z
Miha Brešar, Aleksandar Mijatović
{"title":"Subexponential lower bounds for f-ergodic Markov processes","authors":"Miha Brešar, Aleksandar Mijatović","doi":"10.1007/s00440-024-01306-z","DOIUrl":"https://doi.org/10.1007/s00440-024-01306-z","url":null,"abstract":"<p>We provide a criterion for establishing lower bounds on the rate of convergence in <i>f</i>-variation of a continuous-time ergodic Markov process to its invariant measure. The criterion consists of novel super- and submartingale conditions for certain functionals of the Markov process. It provides a general approach for proving lower bounds on the tails of the invariant measure and the rate of convergence in <i>f</i>-variation of a Markov process, analogous to the widely used Lyapunov drift conditions for upper bounds. Our key technical innovation produces lower bounds on the tails of the heights and durations of the excursions from bounded sets of a continuous-time Markov process using path-wise arguments. We apply our theory to elliptic diffusions and Lévy-driven stochastic differential equations with known polynomial/stretched exponential upper bounds on their rates of convergence. Our lower bounds match asymptotically the known upper bounds for these classes of models, thus establishing their rate of convergence to stationarity. The generality of the approach suggests that, analogous to the Lyapunov drift conditions for upper bounds, our methods can be expected to find applications in many other settings.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182574","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: 0
Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions 李雅普诺夫条件下非 Lipschitz 系数的 Kolmogorov 方程的正则性保持
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-08-20 DOI: 10.1007/s00440-024-01313-0
Martin Chak
{"title":"Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions","authors":"Martin Chak","doi":"10.1007/s00440-024-01313-0","DOIUrl":"https://doi.org/10.1007/s00440-024-01313-0","url":null,"abstract":"<p>Given global Lipschitz continuity and differentiability of high enough order on the coefficients in Itô’s equation, differentiability of associated semigroups, existence of twice differentiable solutions to Kolmogorov equations and weak convergence rates of order one for numerical approximations are known. In this work and against the counterexamples of Hairer et al. (Ann Probab 43(2):468–527, https://doi.org/10.1214/13-AOP838, 2015), the drift and diffusion coefficients having Lipschitz constants that are <span>(o(log V))</span> and <span>(o(sqrt{log V}))</span> respectively for a function <i>V</i> satisfying <span>((partial _t + L)Vle CV)</span> is shown to be a generalizing condition in place of global Lipschitz continuity for the above.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182573","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: 0
The replica-symmetric free energy for Ising spin glasses with orthogonally invariant couplings 具有正交不变耦合的伊辛自旋玻璃的复制对称自由能
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-08-13 DOI: 10.1007/s00440-024-01309-w
Zhou Fan, Yihong Wu
{"title":"The replica-symmetric free energy for Ising spin glasses with orthogonally invariant couplings","authors":"Zhou Fan, Yihong Wu","doi":"10.1007/s00440-024-01309-w","DOIUrl":"https://doi.org/10.1007/s00440-024-01309-w","url":null,"abstract":"<p>We study a variant of the Sherrington–Kirkpatrick (S–K) spin glass model with external field, where the random symmetric couplings matrix does not consist of i.i.d. entries but is instead orthogonally invariant in law. For sufficiently high temperature, we prove a replica-symmetric formula for the first-order limit of the model free energy. Our analysis is an adaptation of a conditional second-moment-method argument previously introduced by Bolthausen for studying the high-temperature regime of the S–K model, where one conditions on the iterates of an Approximate Message Passing (AMP) algorithm for solving the TAP equations for the model magnetization. We apply this method using a memory-free version of AMP that is tailored to the orthogonally invariant structure of the model couplings.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182575","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: 0
On invariant distributions of Feller Markov chains with applications to dynamical systems with random switching 论费勒马尔可夫链的不变分布及其在随机切换动力系统中的应用
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-08-12 DOI: 10.1007/s00440-024-01307-y
Michel Benaïm, Oliver Tough
{"title":"On invariant distributions of Feller Markov chains with applications to dynamical systems with random switching","authors":"Michel Benaïm, Oliver Tough","doi":"10.1007/s00440-024-01307-y","DOIUrl":"https://doi.org/10.1007/s00440-024-01307-y","url":null,"abstract":"<p>We introduce simple conditions ensuring that invariant distributions of a Feller Markov chain on a compact Riemannian manifold are absolutely continuous with a lower semi-continuous, continuous or smooth density with respect to the Riemannian measure. This is applied to Markov chains obtained by random composition of maps and to piecewise deterministic Markov processes obtained by random switching between flows.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182577","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: 0
Martingale-driven integrals and singular SPDEs 马丁格尔驱动积分和奇异 SPDEs
IF 2 1区 数学
Probability Theory and Related Fields Pub Date : 2024-08-12 DOI: 10.1007/s00440-024-01311-2
P. Grazieschi, K. Matetski, H. Weber
{"title":"Martingale-driven integrals and singular SPDEs","authors":"P. Grazieschi, K. Matetski, H. Weber","doi":"10.1007/s00440-024-01311-2","DOIUrl":"https://doi.org/10.1007/s00440-024-01311-2","url":null,"abstract":"<p>We consider multiple stochastic integrals with respect to càdlàg martingales, which approximate a cylindrical Wiener process. We define a chaos expansion, analogous to the case of multiple Wiener stochastic integrals, for these integrals and use it to show moment bounds. Key tools include an iteration of the Burkholder–Davis–Gundy inequality and a multi-scale decomposition similar to the one developed in Hairer and Quastel (Forum Math Pi 6:e3, 2018). Our method can be combined with the recently developed discretisation framework for regularity structures (Hairer and Matetski in Ann Probab 46(3):1651–1709, 2018, Erhard and Hairer in Ann Inst Henri Poincaré Probab Stat 55(4):2209–2248, 2019) to prove convergence of interacting particle systems to singular stochastic PDEs. A companion article (Grazieschiet al. in The dynamical Ising–Kac model in 3D converges to <span>(Phi ^4_3)</span>, 2023. arXiv:2303.10242) applies the results of this paper to prove convergence of a rescaled Glauber dynamics for the three-dimensional Ising–Kac model near criticality to the <span>(Phi ^4_3)</span> dynamics on a torus.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182576","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: 0
The Allen–Cahn equation with weakly critical random initial datum 具有弱临界随机初始基准的艾伦-卡恩方程
IF 1.5 1区 数学
Probability Theory and Related Fields Pub Date : 2024-08-09 DOI: 10.1007/s00440-024-01312-1
Simon Gabriel, Tommaso Rosati, Nikos Zygouras
{"title":"The Allen–Cahn equation with weakly critical random initial datum","authors":"Simon Gabriel, Tommaso Rosati, Nikos Zygouras","doi":"10.1007/s00440-024-01312-1","DOIUrl":"https://doi.org/10.1007/s00440-024-01312-1","url":null,"abstract":"","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141922113","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: 0
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