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Asymptotic Expansions for Additive Measures of Branching Brownian Motions 分支布朗运动加法量的渐近展开
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-19 DOI: 10.1007/s10959-024-01347-z
Haojie Hou, Yan-Xia Ren, Renming Song
{"title":"Asymptotic Expansions for Additive Measures of Branching Brownian Motions","authors":"Haojie Hou, Yan-Xia Ren, Renming Song","doi":"10.1007/s10959-024-01347-z","DOIUrl":"https://doi.org/10.1007/s10959-024-01347-z","url":null,"abstract":"<p>Let <i>N</i>(<i>t</i>) be the collection of particles alive at time <i>t</i> in a branching Brownian motion in <span>(mathbb {R}^d)</span>, and for <span>(uin N(t))</span>, let <span>({textbf{X}}_u(t))</span> be the position of particle <i>u</i> at time <i>t</i>. For <span>(theta in mathbb {R}^d)</span>, we define the additive measures of the branching Brownian motion by </p><span>$$begin{aligned}{} &amp; {} mu _t^theta (textrm{d}{textbf{x}}):= e^{-(1+frac{Vert theta Vert ^2}{2})t}sum _{uin N(t)} e^{-theta cdot {textbf{X}}_u(t)} delta _{left( {textbf{X}}_u(t)+theta tright) }(textrm{d}{textbf{x}}),{} &amp; {} quad textrm{here},, Vert theta Vert mathrm {is, the, Euclidean, norm, of},, theta . end{aligned}$$</span><p>In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for <span>(mu _t^theta (({textbf{a}}, {textbf{b}}]))</span> and <span>(mu _t^theta ((-infty , {textbf{a}}]))</span> for <span>(theta in mathbb {R}^d)</span> with <span>(Vert theta Vert &lt;sqrt{2})</span>, where <span>((textbf{a}, textbf{b}]:=(a_1, b_1]times cdots times (a_d, b_d])</span> and <span>((-infty , textbf{a}]:=(-infty , a_1]times cdots times (-infty , a_d])</span> for <span>(textbf{a}=(a_1,cdots , a_d))</span> and <span>(textbf{b}=(b_1,cdots , b_d))</span>. These expansions sharpen the asymptotic results of Asmussen and Kaplan (Stoch Process Appl 4(1):1–13, 1976) and Kang (J Korean Math Soc 36(1): 139–157, 1999) and are analogs of the expansions in Gao and Liu (Sci China Math 64(12):2759–2774, 2021) and Révész et al. (J Appl Probab 42(4):1081–1094, 2005) for branching Wiener processes (a particular class of branching random walks) corresponding to <span>(theta ={textbf{0}})</span>.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527796","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
Harnack Inequality for Distribution Dependent Second-Order Stochastic Differential Equations 分布相关二阶随机微分方程的哈纳克不等式
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-18 DOI: 10.1007/s10959-024-01346-0
Xing Huang, Xiaochen Ma
{"title":"Harnack Inequality for Distribution Dependent Second-Order Stochastic Differential Equations","authors":"Xing Huang, Xiaochen Ma","doi":"10.1007/s10959-024-01346-0","DOIUrl":"https://doi.org/10.1007/s10959-024-01346-0","url":null,"abstract":"<p>By investigating the regularity of the nonlinear semigroup <span>(P_t^*)</span> associated with the distribution dependent second-order stochastic differential equations, the Harnack inequality is derived when the drift is Lipschitz continuous in the measure variable under the distance induced by the functions being <span>(beta )</span>-Hölder continuous (with <span>(beta &gt; frac{2}{3})</span>) on the degenerate component and square root of Dini continuous on the non-degenerate one. The results extend the existing ones in which the drift is Lipschitz continuous in <span>(L^2)</span>-Wasserstein distance.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"37 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527717","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
Large Deviation Principle for Stochastic Reaction–Diffusion Equations with Superlinear Drift on $$mathbb {R}$$ Driven by Space–Time White Noise 时空白噪声驱动 $$mathbb {R}$ 上超线性漂移的随机反应-扩散方程的大偏差原理
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-17 DOI: 10.1007/s10959-024-01345-1
Yue Li, Shijie Shang, Jianliang Zhai
{"title":"Large Deviation Principle for Stochastic Reaction–Diffusion Equations with Superlinear Drift on $$mathbb {R}$$ Driven by Space–Time White Noise","authors":"Yue Li, Shijie Shang, Jianliang Zhai","doi":"10.1007/s10959-024-01345-1","DOIUrl":"https://doi.org/10.1007/s10959-024-01345-1","url":null,"abstract":"<p>In this paper, we consider stochastic reaction–diffusion equations with superlinear drift on the real line <span>(mathbb {R})</span> driven by space–time white noise. A Freidlin–Wentzell large deviation principle is established by a modified weak convergence method on the space <span>(C([0,T], C_textrm{tem}(mathbb {R})))</span>, where <span>(C_textrm{tem}(mathbb {R}):={fin C(mathbb {R}): sup _{xin mathbb {R}} left( |f(x)|e^{-lambda |x|}right) &lt;infty text { for any } lambda &gt;0})</span>. Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space–time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the specially designed family of norms on the Fréchet space <span>(C([0,T], C_textrm{tem}(mathbb {R})))</span>, one-order moment estimates of the stochastic convolution, and two nonlinear Gronwall-type inequalities play an important role.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529070","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
Hitting with Probability One for Stochastic Heat Equations with Additive Noise 带有加性噪声的随机热方程的概率一命中率
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-06 DOI: 10.1007/s10959-024-01342-4
Robert C. Dalang, Fei Pu
{"title":"Hitting with Probability One for Stochastic Heat Equations with Additive Noise","authors":"Robert C. Dalang, Fei Pu","doi":"10.1007/s10959-024-01342-4","DOIUrl":"https://doi.org/10.1007/s10959-024-01342-4","url":null,"abstract":"<p>We study the hitting probabilities of the solution to a system of <i>d</i> stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive <span>((d-6))</span>-dimensional capacity, the solution visits this set almost surely.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"44 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551865","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
Density Fluctuations for the Multi-Species Stirring Process 多物种搅拌过程的密度波动
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-05-28 DOI: 10.1007/s10959-024-01340-6
Francesco Casini, Cristian Giardinà, Frank Redig
{"title":"Density Fluctuations for the Multi-Species Stirring Process","authors":"Francesco Casini, Cristian Giardinà, Frank Redig","doi":"10.1007/s10959-024-01340-6","DOIUrl":"https://doi.org/10.1007/s10959-024-01340-6","url":null,"abstract":"<p>We study the density fluctuations at equilibrium of the multi-species stirring process, a natural multi-type generalization of the symmetric (partial) exclusion process. In the diffusive scaling limit, the resulting process is a system of infinite-dimensional Ornstein–Uhlenbeck processes that are coupled in the noise terms. This shows that at the level of equilibrium fluctuations the species start to interact, even though at the level of the hydrodynamic limit each species diffuses separately. We consider also a generalization to a multi-species stirring process with a linear reaction term arising from species mutation. The general techniques used in the proof are based on the Dynkin martingale approach, combined with duality for the computation of the covariances.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"63 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166853","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
Explosion Rates for Continuous-State Branching Processes in a Lévy Environment 列维环境下连续状态分支过程的爆炸率
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-05-24 DOI: 10.1007/s10959-024-01338-0
Natalia Cardona-Tobón, Juan Carlos Pardo
{"title":"Explosion Rates for Continuous-State Branching Processes in a Lévy Environment","authors":"Natalia Cardona-Tobón, Juan Carlos Pardo","doi":"10.1007/s10959-024-01338-0","DOIUrl":"https://doi.org/10.1007/s10959-024-01338-0","url":null,"abstract":"<p>Here, we study the long-term behaviour of the non-explosion probability for continuous-state branching processes in a Lévy environment when the branching mechanism is given by the negative of the Laplace exponent of a subordinator. In order to do so, we study the law of this family of processes in the infinite mean case and provide necessary and sufficient conditions for the process to be conservative, i.e. that the process does not explode in finite time a.s. In addition, we establish precise rates for the non-explosion probabilities in the subcritical and critical regimes, first found by Palau et al. (ALEA Lat Am J Probab Math Stat 13(2):1235–1258, 2016) in the case when the branching mechanism is given by the negative of the Laplace exponent of a stable subordinator.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147115","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
A Conditioned Local Limit Theorem for Nonnegative Random Matrices 非负随机矩阵的条件局部极限定理
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-05-13 DOI: 10.1007/s10959-024-01336-2
Marc Peigné, Da Cam Pham
{"title":"A Conditioned Local Limit Theorem for Nonnegative Random Matrices","authors":"Marc Peigné, Da Cam Pham","doi":"10.1007/s10959-024-01336-2","DOIUrl":"https://doi.org/10.1007/s10959-024-01336-2","url":null,"abstract":"<p>For any fixed real <span>(a &gt; 0)</span> and <span>(x in {mathbb {R}}^d, d ge 1)</span>, we consider the real-valued random process <span>((S_n)_{n ge 0})</span> defined by <span>( S_0= a, S_n= a+ln vert g_ncdots g_1xvert , n ge 1)</span>, where the <span>(g_k, k ge 1, )</span> are i.i.d. nonnegative random matrices. By using the strategy initiated by Denisov and Wachtel to control fluctuations in cones of <i>d</i>-dimensional random walks, we obtain an asymptotic estimate and bounds on the probability that the process <span>((S_n)_{n ge 0})</span> remains nonnegative up to time <i>n</i> and simultaneously belongs to some compact set <span>([b, b+ell ]subset {mathbb {R}}^+_*)</span> at time <i>n</i>.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"61 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931528","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
Measure Pseudo-S-asymptotically Bloch-Type Periodicity of Some Semilinear Stochastic Integrodifferential Equations 测量某些半线性随机积分微分方程的伪 S-渐近布洛赫型周期性
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-05-03 DOI: 10.1007/s10959-024-01335-3
Amadou Diop, Mamadou Moustapha Mbaye, Yong-Kui Chang, Gaston Mandata N’Guérékata
{"title":"Measure Pseudo-S-asymptotically Bloch-Type Periodicity of Some Semilinear Stochastic Integrodifferential Equations","authors":"Amadou Diop, Mamadou Moustapha Mbaye, Yong-Kui Chang, Gaston Mandata N’Guérékata","doi":"10.1007/s10959-024-01335-3","DOIUrl":"https://doi.org/10.1007/s10959-024-01335-3","url":null,"abstract":"<p>This paper gives a new property for stochastic processes, called square-mean <span>(mu -)</span>pseudo-<i>S</i>-asymptotically Bloch-type periodicity. We show how this property is preserved under some operations, such as composition and convolution, for stochastic processes. Our main results extend the classical results on S-asymptotically Bloch-type periodic functions. We also apply our results to some problems involving semilinear stochastic integrodifferential equations in abstract spaces</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"169 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887391","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
Multi-dimensional Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion with Diagonal Generators 带对角线发生器的 G 布朗运动驱动的多维反射后向随机微分方程
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-04-26 DOI: 10.1007/s10959-024-01334-4
Hanwu Li, Guomin Liu
{"title":"Multi-dimensional Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion with Diagonal Generators","authors":"Hanwu Li, Guomin Liu","doi":"10.1007/s10959-024-01334-4","DOIUrl":"https://doi.org/10.1007/s10959-024-01334-4","url":null,"abstract":"<p>We consider the well-posedness problem of multi-dimensional reflected backward stochastic differential equations driven by <i>G</i>-Brownian motion (<i>G</i>-BSDEs) with diagonal generators. Two methods, including the penalization method and the Picard iteration argument, are provided to prove the existence and uniqueness of the solutions. We also study its connection with the obstacle problem of a system of fully nonlinear PDEs.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805674","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 the $$L^{p}$$ -Spaces of Projective Limits of Probability Measures 论概率措施的投影极限的 $$L^{p}$$ 空间
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-04-20 DOI: 10.1007/s10959-024-01329-1
Juan Carlos Sampedro
{"title":"On the $$L^{p}$$ -Spaces of Projective Limits of Probability Measures","authors":"Juan Carlos Sampedro","doi":"10.1007/s10959-024-01329-1","DOIUrl":"https://doi.org/10.1007/s10959-024-01329-1","url":null,"abstract":"<p>The present article describes the precise structure of the <span>(L^{p})</span>-spaces of projective limit measures by introducing a category-theoretical perspective. This analysis is applied to measures on vector spaces and in particular to Gaussian measures on nuclear topological vector spaces. A simple application to constructive quantum field theory (QFT) is given through the Osterwalder–Schrader axioms.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"3 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140635161","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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