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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=666210
• 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 proposed*research 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-Dirichlet*form 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 we*can completely solve Getoor's conjecture and give an explicit criterion on the validity of (H) for general Markov*processes.******2) Construction of the two-parameter Fleming-Viot process. The two-parameter Dirichlet process has a lot of*applications in mathematical population genetics and Bayesian nonparametric statistics. Through the efforts of many researchers, people*now have good understanding of its various properties. However, people still do not know much about its*associated dynamic model. Construction of the two-parameter Fleming-Viot process with a general state space is a challenging open*problem in the area of combinatorial probability. We expect to solve this problem based on our own work*and other references published in recent years.******3) Large deviations for non-symmetric*Markov processes. Takeda and his collaborators have systematically developed the Donsker-Varadhan*type large deviation principle for time reversible Markov processes. However, not many results have been obtained for the non-symmetric*case. By virtue of recent results on*stochastic calculus of Markov processes associated with semi-Dirichlet forms and generalized Feynman-Kac semigroups, we expect to obtain the large*deviation principle for the occupation time distributions of general non-symmetric*Markov processes with generalized Feynman-Kac functionals and extend some remarkable results of Takeda's group to the framework of*semi-Dirichlet*forms. ******4) Boundary value problems with non-local operators and singular nonlinearities. In recent years, people have used probabilistic approach to study various boundary value*problems. In this project, we will use the Dirichlet form theory to consider the boundary value problem for a*very 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。本提议的*研究计划致力于进行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，即基本上所有 Levy 过程都满足*(H) 仍然没有得到解决。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，并给出一个明确的标准（H）对于一般马尔可夫*过程的有效性。******2）两个的构造-参数弗莱明-维奥特过程。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很多应用。经过许多研究人员的努力，人们现在对其各种特性有了很好的了解。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。 Takeda 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费曼-Kac半群相关的马尔可夫过程随机微积分的最新结果，我们期望获得广义费曼非对称马尔可夫过程的占用时间分布的大偏差原理-Kac泛函并将Takeda小组的一些显着成果扩展到*半狄利克雷*形式的框架。 ******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一类非常一般的非对称和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、唯一性和规律性，以及解的概率表示。**