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.

狄利克雷形式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架起了一座桥梁，效益双向流动。本研究计划致力于狄利克雷形式和相关随机分析的理论研究。我们将重点关注四个重要问题：1）亨特假设（H）和格托尔猜想。亨特假设（H）在概率势论中起着至关重要的作用。众所周知，任何与半狄利克雷形式相关的马尔可夫过程本质上都满足（H）。然而，文献中缺乏关于 (H) 对于一般马尔可夫过程的有效性的有力表征。特别是，Getoor 的猜想（基本上所有 Levy 过程都满足（H））仍未得到解决。基于我们近年来发表的论文，我们希望能够彻底解决Getoor猜想，并对一般马尔可夫过程的(H)的有效性给出明确的判据。2)二参数Fleming-Viot过程的构造。双参数狄利克雷过程在数学群体遗传学和贝叶斯非参数统计中有大量应用。经过许多研究人员的努力，人们现在对其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仍然知之甚少。构造具有一般状态空间的二参数 Fleming-Viot 过程是组合概率领域中一个具有挑战性的开放问题。我们希望根据我们自己的工作和近年来发表的其他参考文献来解决这个问题。3）非对称马尔可夫过程的大偏差。武田和他的合作者系统地开发了时间可逆马尔可夫过程的 Donsker-Varadhan 型大偏差原理。然而，对于非对称情况，获得的结果并不多。凭借与半狄利克雷形式和广义Feynman-Kac半群相关的马尔可夫过程的随机微积分的最新成果，我们期望获得具有广义Feynman-Kac泛函的一般非对称马尔可夫过程的占用时间分布的大偏差原理，并推广一些Takeda团队在半狄利克雷形式框架上取得了显着的成果。 4) 非局部算子和奇异非线性的边值问题。近年来，人们利用概率方法来研究各种边值问题。在本项目中，我们将使用狄利克雷形式理论来考虑具有奇异非线性的非对称和非局部算子的一般类别的边值问题。我们期望建立边值问题解的存在性、唯一性和规律性以及解的概率表示。