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。这个拟议的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。 首先，我们将进一步发展狄利克雷形式的理论。我们将研究Hunt假设（H）与非对称Dirichlet型的扇形条件之间的关系。特别是，我们将集中在Getoor的猜想一般Levy过程和最近的猜想，右（强马尔可夫）过程满足亨特的假设（H）当且仅当它是局部相关的扇形Dirichlet形式。我们期望扩展现有的非对称Dirichlet型的框架，并大大拓宽Dirichlet型理论的应用。我们将研究与非对称（半）狄利克雷形式相关的马尔可夫过程的随机微积分。此外，我们还将相关结果应用于非对称Markov过程的乘性泛函、奇异系数Dirichlet和混合边值问题、非对称Markov过程的LDP和L^p独立性等问题的研究。 其次，我们将应用狄利克雷形式的理论对随机分析的几个重要问题。a）我们将使用Dirichlet形式作为工具来考虑一般双参数Fleming-Viot过程的构造。我们期望在数学群体遗传学领域彻底解决这一重要的开放问题。B）我们将应用Dirichlet形式理论对奇异/无限维信号进行非线性滤波，并研究滤波方程解的各种数值近似。在这项工作中开发的算法有可能被应用到加拿大的工业和军事。（3）利用Dirichlet形式系统地研究具有奇异系数的倒向随机微分方程。所得结果可应用于金融数学中的某些问题。

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.