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Regularity of the Semigroup of Regular Probability Measures on Locally Compact Hausdorff Topological Groups 局部紧密豪斯多夫拓扑群上规则概率量半群的规则性
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-07-02 DOI: 10.1007/s10959-024-01353-1
M. N. N. Namboodiri
{"title":"Regularity of the Semigroup of Regular Probability Measures on Locally Compact Hausdorff Topological Groups","authors":"M. N. N. Namboodiri","doi":"10.1007/s10959-024-01353-1","DOIUrl":"https://doi.org/10.1007/s10959-024-01353-1","url":null,"abstract":"<p>Let <i>G</i> be a locally compact Hausdorff group, and let <i>P</i>(<i>G</i>) denote the class of all regular probability measures on <i>G</i>. It is well known that <i>P</i>(<i>G</i>) forms a semigroup under the convolution of measures. In this paper, we prove that <i>P</i>(<i>G</i>) is not algebraically regular in the sense that not every element has a generalized inverse. Additionally, we attempt to identify algebraically regular elements in some exceptional cases. Several supporting examples are provided to justify these assumptions.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503844","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
Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions 一类高斯过程的 Kac-Stroock 近似函数大偏差与小噪声扩散的应用
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-27 DOI: 10.1007/s10959-024-01354-0
Jiang Hui, Xu Lihu, Yang Qingshan
{"title":"Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions","authors":"Jiang Hui, Xu Lihu, Yang Qingshan","doi":"10.1007/s10959-024-01354-0","DOIUrl":"https://doi.org/10.1007/s10959-024-01354-0","url":null,"abstract":"<p>In this paper, we establish the functional large deviation principle (LDP) for the Kac–Stroock approximations of a wild class of Gaussian processes constructed by telegraph types of integrals with <span>(L^2)</span>-integrands under mild conditions and find the explicit form for their rate functions. Our investigation includes a broad range of kernels, such as those related to Brownian motions, fractional Brownian motions with whole Hurst parameters, and Ornstein–Uhlenbeck processes. Furthermore, we consider a class of non-Markovian stochastic differential equations driven by the Kac–Stroock approximation and establish their Freidlin–Wentzell type LDP. The rate function clearly indicates an interesting phase transition phenomenon as the approximating rate moves from one region to the other.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"158 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503845","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
Sample Path Behaviors of Lévy Processes Conditioned to Avoid Zero 以避零为条件的莱维过程的路径行为样本
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-26 DOI: 10.1007/s10959-024-01352-2
Shosei Takeda
{"title":"Sample Path Behaviors of Lévy Processes Conditioned to Avoid Zero","authors":"Shosei Takeda","doi":"10.1007/s10959-024-01352-2","DOIUrl":"https://doi.org/10.1007/s10959-024-01352-2","url":null,"abstract":"<p>Takeda and Yano (Electron J Probab 28:1–35, 2023) determined the limit of Lévy processes conditioned to avoid zero via various random clocks in terms of Doob’s <span>(h)</span>-transform, where the limit processes may differ according to the choice of random clocks. The purpose of this paper is to investigate sample path behaviors of the limit processes in long time and in short time.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"74 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503846","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 Generalized Birth–Death Process and Its Linear Versions 论广义出生-死亡过程及其线性版本
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-22 DOI: 10.1007/s10959-024-01355-z
P. Vishwakarma, K. K. Kataria
{"title":"On the Generalized Birth–Death Process and Its Linear Versions","authors":"P. Vishwakarma, K. K. Kataria","doi":"10.1007/s10959-024-01355-z","DOIUrl":"https://doi.org/10.1007/s10959-024-01355-z","url":null,"abstract":"<p>In this paper, we consider a generalized birth–death process (GBDP) and examine its linear versions. Using its transition probabilities, we obtain the system of differential equations that governs its state probabilities. The distribution function of its waiting time in state <i>s</i> given that it starts in state <i>s</i> is obtained. For a linear version of it, namely the generalized linear birth–death process (GLBDP), we obtain the probability generating function, mean, variance and the probability of ultimate extinction of population. Also, we obtain the maximum likelihood estimate of its parameters. The differential equations that govern the joint cumulant generating functions of the population size with cumulative births and cumulative deaths are derived. In the case of constant birth and death rates in GBDP, the explicit forms of the state probabilities, joint probability mass functions of population size with cumulative births and cumulative deaths, and their marginal probability mass functions are obtained. It is shown that the Laplace transform of an integral of GBDP satisfies its Kolmogorov backward equation with certain scaled parameters. The first two moments of the path integral of GLBDP are obtained. Also, we consider the immigration effect in GLBDP for two different cases. An application of a linear version of GBDP and its path integral to a vehicles parking management system is discussed. Later, we introduce a time-changed version of the GBDP where time is changed via an inverse stable subordinator. We show that its state probabilities are governed by a system of fractional differential equations.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"11 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527795","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
Some Families of Random Fields Related to Multiparameter Lévy Processes 与多参数莱维过程有关的一些随机场族
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-21 DOI: 10.1007/s10959-024-01351-3
Francesco Iafrate, Costantino Ricciuti
{"title":"Some Families of Random Fields Related to Multiparameter Lévy Processes","authors":"Francesco Iafrate, Costantino Ricciuti","doi":"10.1007/s10959-024-01351-3","DOIUrl":"https://doi.org/10.1007/s10959-024-01351-3","url":null,"abstract":"<p>Let <span>({mathbb {R}}^N_+= [0,infty )^N)</span>. We here make new contributions concerning a class of random fields <span>((X_t)_{tin {mathbb {R}}^N_+})</span> which are known as multiparameter Lévy processes. Related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators. We also provide a Phillips formula concerning the composition of <span>((X_t)_{tin {mathbb {R}}^N_+})</span> by means of subordinator fields. We finally define the composition of <span>((X_t)_{tin {mathbb {R}}^N_+})</span> by means of the so-called inverse random fields, which gives rise to interesting long-range dependence properties. As a byproduct of our analysis, we present a model of anomalous diffusion in an anisotropic medium which extends the one treated in Beghin et al. (Stoch Proc Appl 130:6364–6387, 2020), by improving some of its shortcomings.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503847","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
Determining the Number and Values of Thresholds for Multi-regime Threshold Ornstein–Uhlenbeck Processes 确定多制度阈值奥恩斯坦-乌伦贝克过程的阈值数量和数值
IF 0.8 4区 数学
Journal of Theoretical Probability Pub Date : 2024-06-21 DOI: 10.1007/s10959-024-01343-3
Dingwen Zhang
{"title":"Determining the Number and Values of Thresholds for Multi-regime Threshold Ornstein–Uhlenbeck Processes","authors":"Dingwen Zhang","doi":"10.1007/s10959-024-01343-3","DOIUrl":"https://doi.org/10.1007/s10959-024-01343-3","url":null,"abstract":"<p>The threshold Ornstein–Uhlenbeck process is a stochastic process that satisfies a stochastic differential equation with a drift term modeled as a piecewise linear function and a diffusion term characterized by a positive constant. This paper addresses the challenge of determining both the number and values of thresholds based on the continuously observed process. We present three testing algorithms aimed at determining the unknown number and values of thresholds, in conjunction with least squares estimators for drift parameters. The limiting distribution of the proposed test statistic is derived. Additionally, we employ sequential and global methods to determine the values of thresholds, and prove their weak convergence. Monte Carlo simulation results are provided to illustrate and support our theoretical findings. We utilize the model to estimate the term structure of US treasury rates and currency foreign exchange rates.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"11 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528924","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
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
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