Dirichlet Forms and Stochastic Analysis

狄利克雷形式和随机分析

• 批准号: RGPIN-2018-04394
• 负责人: Sun, Wei
• 金额: $ 1.46万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759137
• 关键词: Dirichlet; Forms; Stochastic; Analysis

The Dirichlet form theory is one of the most active areas of modern probability theory and stochastic processes.It establishes a bridge between analysis and probability, and the benefits flow in both directions. This proposedresearch program is devoted to performing theoretical research in Dirichlet forms and related stochastic analysis.We will focus on four important problems.1) Hunt's hypothesis (H) and Getoor's conjecture. Hunt's hypothesis (H) plays a crucial role in probabilistic potential theory. It is well-known that any Markov process associated with a semi-Dirichletform essentially satisfies (H). However, there lacks powerful characterization in literature regarding the validity of(H) for general Markov processes. In particular, Getoor's conjecture that essentially all Levy processes satisfy(H) still remains unsolved. Based on the papers that we published in recent years, we hope wecan completely solve Getoor's conjecture and give an explicit criterion on the validity of (H) for general Markovprocesses.2) Construction of the two-parameter Fleming-Viot process. The two-parameter Dirichlet process has a lot ofapplications in mathematical population genetics and Bayesian nonparametric statistics. Through the efforts of many researchers, peoplenow have good understanding of its various properties. However, people still do not know much about itsassociated dynamic model. Construction of the two-parameter Fleming-Viot process with a general state space is a challenging openproblem in the area of combinatorial probability. We expect to solve this problem based on our own workand other references published in recent years.3) Large deviations for non-symmetricMarkov processes. Takeda and his collaborators have systematically developed the Donsker-Varadhantype large deviation principle for time reversible Markov processes. However, not many results have been obtained for the non-symmetriccase. By virtue of recent results onstochastic calculus of Markov processes associated with semi-Dirichlet forms and generalized Feynman-Kac semigroups, we expect to obtain the largedeviation principle for the occupation time distributions of general non-symmetricMarkov processes with generalized Feynman-Kac functionals and extend some remarkable results of Takeda's group to the framework ofsemi-Dirichletforms. 4) Boundary value problems with non-local operators and singular nonlinearities. In recent years, people have used probabilistic approach to study various boundary valueproblems. In this project, we will use the Dirichlet form theory to consider the boundary value problem for avery general class of non-symmetric and nonlocal operators with singular nonlinearities. We expect to establish the existence, uniqueness, and regularity of solutions to the boundary value problem as well as the probabilistic representation of the solutions.

狄利克雷型理论是现代概率论和随机过程最活跃的领域之一，它在分析和概率之间建立了一座桥梁，利益双向流动。本研究计画将致力于狄利克雷型及相关随机分析的理论研究，主要探讨四个重要问题：1）亨特假说（H）与盖图尔猜想。Hunt假设（H）在概率势理论中起着至关重要的作用。众所周知，任何与半狄利克雷型相关的马尔可夫过程本质上满足（H）。然而，在文献中缺乏强有力的表征（H）的有效性一般马尔可夫过程。特别是，Getoor的猜想，基本上所有的Levy过程满足（H）仍然没有解决。在此基础上，我们希望能完全解决Getoor猜想，并给出（H）对一般马氏过程有效性的一个明确判据。2）两参数Fleming-Viot过程的构造。双参数Dirichlet过程在数学群体遗传学和贝叶斯非参数统计中有着广泛的应用。经过许多研究者的努力，人们对它的各种性质有了很好的了解。然而，人们对其相关的动力学模型还知之甚少。在一般状态空间中构造两参数Fleming-Viot过程是组合概率领域中一个具有挑战性的开放问题。我们期望在自己的工作和近年来发表的其他文献的基础上解决这个问题。3）非线性马尔可夫过程的大偏差。武田和他的合作者系统地发展了时间可逆马尔可夫过程的Donsker-Varadhan型大偏差原理。然而，没有太多的结果已经获得的非pacliccase。利用半Dirichlet型和广义Feynman-Kac半群的Markov过程的随机微积分的最新结果，我们期望得到广义Feynman-Kac泛函的一般非连续Markov过程的占有时间分布的大偏差原理，并将Takeda群的一些重要结果推广到半Dirichlet型的框架中. 4)具非局部算子及奇异非线性项的边值问题。近年来，人们用概率方法研究了各种边值问题.在这个项目中，我们将使用Dirichlet型理论来考虑一类非常一般的具有奇异非线性项的非对称非局部算子的边值问题。我们期望建立边值问题解的存在性、唯一性和正则性，以及解的概率表示。