Dirichlet Forms and Stochastic Analysis

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，利益在两个方向流动。这一拟议 研究计划致力于在狄利克雷形式和相关的随机分析进行理论研究。 我们将重点关注四个重要问题。 1)Hunt的假设（H）和Getoor的猜想。Hunt假设（H）在概率势理论中起着至关重要的作用。众所周知，任何与半狄利克雷相关的马尔可夫过程 形式基本上满足（H）。然而，在文学作品中缺乏关于有效性的有力描述， (H)一般马尔可夫过程。特别地，Getoor猜想基本上所有Levy过程都满足 (H)仍然没有解决。根据我们近年来发表的论文，我们希望我们 完全解决了Getoor猜想，并给出了一般Markov情形下（H）有效性的一个显式判据 流程. 2)双参数Fleming-Viot过程的构造双参数Dirichlet过程有很多 在数学群体遗传学和贝叶斯非参数统计中的应用。通过许多研究人员的努力， 现在对它的各种特性有了很好的了解。然而，人们对它的了解仍然不多。 相关动态模型构造具有一般状态空间的双参数Fleming-Viot过程是一个具有挑战性的开放性问题。 组合概率领域的问题。我们期望通过自己的工作来解决这个问题 以及近年来发表的其他参考文献。 3)非对称的大偏差 马尔可夫过程武田和他的合作者系统地开发了Donsker-Varadhan 型大偏差原理。然而，对于非对称的， 案子根据最近的研究结果， 随机微积分的马尔可夫过程与半狄利克雷形式和广义Feynman-Kac半群，我们期望获得大的 一般非对称占有时间分布的偏差原理 本文讨论了具有广义Feynman-Kac泛函的Markov过程，并将Takeda群的一些重要结果推广到 半狄利克雷 forms. 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各种边值问题 问题在本项目中，我们将使用狄利克雷形式理论来考虑a的边值问题 具有奇异非线性的非对称和非局部算子的非常一般的类。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。

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.