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The Voter Model with a Slow Membrane 带慢膜的选民模型
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-03-13 DOI: 10.1007/s10959-024-01321-9
Linjie Zhao, Xiaofeng Xue
{"title":"The Voter Model with a Slow Membrane","authors":"Linjie Zhao, Xiaofeng Xue","doi":"10.1007/s10959-024-01321-9","DOIUrl":"https://doi.org/10.1007/s10959-024-01321-9","url":null,"abstract":"<p>We introduce the voter model on the infinite integer lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. The voter model is one of the classical interacting particle systems with state space <span>({0,1}^{mathbb Z^d})</span>. In our model, a voter adopts one of its neighbors’ opinion at rate one except for neighbors crossing the hyperplane <span>({x:x_1 = 1/2})</span>, where the rate is <span>(alpha N^{-beta })</span> and thus is called a slow membrane. Above, <span>(alpha &gt;0 textrm{and} beta ge 0)</span> are given parameters and the positive integer <i>N</i> is a scaling parameter. We consider the limit <span>(N rightarrow infty )</span> and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of <span>(beta )</span>. We also consider the nonequilibrium fluctuations, where the limit is described by generalized Ornstein–Uhlenbeck processes with certain boundary conditions corresponding to the hydrodynamic equation.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"43 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140116330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Rough Differential Equations Containing Path-Dependent Bounded Variation Terms 包含路径依赖性有界变量项的粗糙微分方程
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-03-08 DOI: 10.1007/s10959-024-01319-3
Shigeki Aida
{"title":"Rough Differential Equations Containing Path-Dependent Bounded Variation Terms","authors":"Shigeki Aida","doi":"10.1007/s10959-024-01319-3","DOIUrl":"https://doi.org/10.1007/s10959-024-01319-3","url":null,"abstract":"<p>We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent stochastic differential equations containing running maximum processes and normal reflection terms. We apply these results to determine the topological support of the solution processes.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140075488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Homogenization of a Multivariate Diffusion with Semipermeable Interfaces 带半透界面的多元扩散的均质化
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-03-07 DOI: 10.1007/s10959-024-01317-5
{"title":"Homogenization of a Multivariate Diffusion with Semipermeable Interfaces","authors":"","doi":"10.1007/s10959-024-01317-5","DOIUrl":"https://doi.org/10.1007/s10959-024-01317-5","url":null,"abstract":"<h3>Abstract</h3> <p>We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular <em>interface-induced</em> drift.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140075635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Precise Tail Behaviour of Some Dirichlet Series 某些德里赫利数列的精确尾部行为
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-03-05 DOI: 10.1007/s10959-024-01318-4
Alexander Iksanov, Vitali Wachtel
{"title":"Precise Tail Behaviour of Some Dirichlet Series","authors":"Alexander Iksanov, Vitali Wachtel","doi":"10.1007/s10959-024-01318-4","DOIUrl":"https://doi.org/10.1007/s10959-024-01318-4","url":null,"abstract":"<p>Let <span>(eta _1)</span>, <span>(eta _2,ldots )</span> be independent copies of a random variable <span>(eta )</span> with zero mean and finite variance which is bounded from the right, that is, <span>(eta le b)</span> almost surely for some <span>(b&gt;0)</span>. Considering different types of the asymptotic behaviour of the probability <span>(mathbb {P}{eta in [b-x,b]})</span> as <span>(xrightarrow 0+)</span>, we derive precise tail asymptotics of the random Dirichlet series <span>(sum _{kge 1}k^{-alpha }eta _k)</span> for <span>(alpha in (1/2, 1])</span>.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 Poisson Approximation 关于泊松逼近
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-28 DOI: 10.1007/s10959-023-01310-4
S. Y. Novak
{"title":"On Poisson Approximation","authors":"S. Y. Novak","doi":"10.1007/s10959-023-01310-4","DOIUrl":"https://doi.org/10.1007/s10959-023-01310-4","url":null,"abstract":"<p>The problem of evaluating the accuracy of Poisson approximation to the distribution of a sum of independent integer-valued random variables has attracted a lot of attention in the past six decades. Among authors who contributed to the topic are Prokhorov, Kolmogorov, LeCam, Shorgin, Barbour, Hall, Deheuvels, Pfeifer, Roos, and many others. From a practical point of view, the problem has important applications in insurance, reliability theory, extreme value theory, etc. From a theoretical point of view, the topic provides insights into Kolmogorov’s problem concerning the accuracy of approximation of the distribution of a sum of independent random variables by infinitely divisible laws. The task of establishing an estimate of the accuracy of Poisson approximation with a correct (the best possible) constant at the leading term remained open for decades. We present a solution to that problem in the case where the accuracy of approximation is evaluated in terms of the point metric. We generalise and sharpen the corresponding inequalities established by preceding authors. A new result is established for the intensively studied topic of compound Poisson approximation to the distribution of a sum of integer-valued r.v.s.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces 在双曲空间上时序演化的多变量随机场
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-28 DOI: 10.1007/s10959-024-01316-6
Anatoliy Malyarenko, Emilio Porcu
{"title":"Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces","authors":"Anatoliy Malyarenko, Emilio Porcu","doi":"10.1007/s10959-024-01316-6","DOIUrl":"https://doi.org/10.1007/s10959-024-01316-6","url":null,"abstract":"<p>Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the <i>n</i>-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains 几何尔格马尔可夫链加法函数的概率和矩不等式
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-18 DOI: 10.1007/s10959-024-01315-7
Alain Durmus, Eric Moulines, Alexey Naumov, Sergey Samsonov
{"title":"Probability and Moment Inequalities for Additive Functionals of Geometrically Ergodic Markov Chains","authors":"Alain Durmus, Eric Moulines, Alexey Naumov, Sergey Samsonov","doi":"10.1007/s10959-024-01315-7","DOIUrl":"https://doi.org/10.1007/s10959-024-01315-7","url":null,"abstract":"<p>In this paper, we establish moment and Bernstein-type inequalities for additive functionals of geometrically ergodic Markov chains. These inequalities extend the corresponding inequalities for independent random variables. Our conditions cover Markov chains converging geometrically to the stationary distribution either in weighted total variation norm or in weighted Wasserstein distances. Our inequalities apply to unbounded functions and depend explicitly on constants appearing in the conditions that we consider.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"63 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139902475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term 无漂移项非线性随机热方程的不变度量
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-15 DOI: 10.1007/s10959-023-01302-4
Le Chen, Nicholas Eisenberg
{"title":"Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term","authors":"Le Chen, Nicholas Eisenberg","doi":"10.1007/s10959-023-01302-4","DOIUrl":"https://doi.org/10.1007/s10959-023-01302-4","url":null,"abstract":"<p>This paper deals with the long-term behavior of the solution to the nonlinear stochastic heat equation <span>(frac{partial u}{partial t} - frac{1}{2}Delta u = b(u){dot{W}})</span>, where <i>b</i> is assumed to be a globally Lipschitz continuous function and the noise <span>({dot{W}})</span> is a centered and spatially homogeneous Gaussian noise that is white in time. We identify a set of nearly optimal conditions on the initial data, the correlation measure of the noise, and the weight function <span>(rho )</span>, which together guarantee the existence of an invariant measure in the weighted space <span>(L^2_rho ({mathbb {R}}^d))</span>. In particular, our result covers the <i>parabolic Anderson model</i> (i.e., the case when <span>(b(u) = lambda u)</span>) starting from the Dirac delta measure.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Non-uniqueness Phase of Percolation on Reflection Groups in $${mathbb {H}^3}$$ 反射群在 $${mathbb {H}^3}$ 中的周遍非唯一性阶段
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-15 DOI: 10.1007/s10959-024-01313-9
Jan Czajkowski
{"title":"Non-uniqueness Phase of Percolation on Reflection Groups in $${mathbb {H}^3}$$","authors":"Jan Czajkowski","doi":"10.1007/s10959-024-01313-9","DOIUrl":"https://doi.org/10.1007/s10959-024-01313-9","url":null,"abstract":"<p>We consider Bernoulli bond and site percolation on Cayley graphs of reflection groups in the three-dimensional hyperbolic space <span>({mathbb {H}^3})</span> corresponding to a very large class of Coxeter polyhedra. In such setting, we prove the existence of a non-empty non-uniqueness percolation phase, i.e. that <span>(p_c &lt; p_u)</span>. This means that for some values of the Bernoulli percolation parameter there are a.s. infinitely many infinite components in the percolation subgraph. The proof relies on upper estimates for the spectral radius of the graph and on a lower estimate for its growth rate. The latter estimate involves only the number of generators of the group and is proved in the article as well.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Fractional Skellam Process of Order k k 阶分数斯凯拉姆过程
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-02-12 DOI: 10.1007/s10959-024-01314-8
{"title":"Fractional Skellam Process of Order k","authors":"","doi":"10.1007/s10959-024-01314-8","DOIUrl":"https://doi.org/10.1007/s10959-024-01314-8","url":null,"abstract":"<h3>Abstract</h3> <p>We introduce and study a fractional version of the Skellam process of order <em>k</em> by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order <em>k</em> (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of FSPoK which are utilized to establish its long-range dependence property. Later, we consider two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes. </p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"101 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139770316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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