Dirichlet Forms and Stochastic Analysis

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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.