Dirichlet Forms and Stochastic Analysis

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The theory of Dirichlet forms 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. First, we will further develop the theory of Dirichlet forms. We will investigate the relationship between Hunt's hypothesis (H) and the sector condition of non-symmetric Dirichlet forms. In particular, we will concentrate on Getoor's conjecture for general Levy processes and the recent conjecture that a right (strong Markov) process satisfies Hunt's hypothesis (H) if and only if it is locally associated with a sectorial Dirichlet form. We expect to extend the existing framework of non-symmetric Dirichlet forms and largely broaden the applications of the theory of Dirichlet forms. We will study the stochastic calculus of Markov processes associated with non-symmetric (semi)-Dirichlet forms. Moreover, we will apply the related results to the study of multiplicative functionals of non-symmetric Markov processes, Dirichlet and mixed boundary-valued problems with singular coefficients, LDP and L^p independence of non-symmetric Markov processes, etc. Second, we will apply the theory of Dirichlet forms to several important problems of stochastic analysis. a) We will use Dirichlet forms as a tool to consider the construction of general two-parameter Fleming-Viot processes. We expect to completely solve this important open problem in the area of mathematical population genetics. b) We will apply the theory of Dirichlet forms to nonlinear filtering of singular/infinite-dimensional signals and study various numerical approximations to solutions of the filtering equations. The algorithms developed in this work have the potential to be applied to Canadian industry and military. c) We will use Dirichlet forms to systematically study backward stochastic differential equations with singular coefficients. The obtained results can be applied to some problems of mathematical finance.