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。这个建议 * 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 * 形式相关联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。基于我们近年来发表的论文，我们希望能彻底解决Getoor猜想，并给出一般Markov过程（H）有效性的一个明确的判据。2)双参数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。通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。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，对于非对称情形，得到的结果并不多。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泛函的一般非对称Markov过程占据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型的框架下. **4）具有非局部算子和奇异非线性项的边值问题。近年来，人们开始用概率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非对称非局部奇异非线性算子的边值问题。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。

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.**