Dirichlet Forms and Stochastic Analysis

狄利克雷型理论是现代概率论和随机过程中最活跃的领域之一。它在分析和概率之间建立了一座桥梁，收益是双向流动的。这个拟议的研究方案致力于进行狄利克雷形式的理论研究和相关的随机分析。*我们将集中在四个重要问题上。*1)亨特假说(H)和盖托猜想。亨特假设(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Dirichlet*形式相关的马尔可夫过程本质上满足(H)。然而，文献中对一般马氏过程*(H)的有效性缺乏有力的刻画。具体地说，GToor关于本质上所有的Levy过程都满足*(H)的猜想仍未解决。基于我们近几年发表的论文，我们希望能够完全解决GToor猜想，并给出一般马氏过程(H)有效性的一个显式判据。两参数Dirichlet过程在数学种群遗传学和贝叶斯非参数统计中有着广泛的应用。经过许多研究人员的努力，人们现在对它的各种性质有了很好的了解。然而，人们对其相关的动态模型仍然知之甚少。构造具有一般状态空间的两参数Fleming-Viot过程是组合概率领域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)非对称马尔可夫过程的大偏差。武田及其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，对于非对称情况，并没有得到很多结果。借助于与半Dirichlet形式和广义Feynman-Kac半群相关的马尔可夫过程的*随机演算的最新结果，我们期望得到具有广义Feynman-Kac泛函的一般非对称*马氏过程的占用时间分布的大偏差原理，并将Takeda群的一些显著结果推广到*半Dirichlet*形式的框架下.*4)具有非局部算子和奇异非线性的边值问题。近年来，人们用概率论的方法研究各种边值问题。在这个项目中，我们将使用Dirichlet形式理论来考虑一类非常一般的具有奇异非线性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.**