Dirichlet Forms and Stochastic Analysis

狄利克雷形式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架起了一座桥梁，而且效益是双向流动的。本研究计划致力于狄利克雷形式的理论研究及相关的随机分析。首先，我们将进一步发展狄利克雷形式理论。我们将研究Hunt假设(H)与非对称Dirichlet形式扇形条件之间的关系。特别地，我们将集中在一般Levy过程的Getoor猜想和最近的猜想，即一个右（强马尔可夫）过程满足Hunt的假设(H)当且仅当它局部与一个扇形Dirichlet形式相关联。我们期望扩展非对称狄利克雷形式的现有框架，并极大地拓宽狄利克雷形式理论的应用。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演算。此外，我们将把相关结果应用于非对称马尔可夫过程的乘法泛函、Dirichlet和奇异系数混合边值问题、非对称马尔可夫过程的LDP和L^p独立性等方面的研究。其次，我们将把狄利克雷形式理论应用于随机分析的几个重要问题。a)我们将使用狄利克雷形式作为工具来考虑一般双参数Fleming-Viot过程的构造。我们期望彻底解决这一数学种群遗传学领域的重要开放性问题。b)我们将把狄利克雷形式理论应用于奇异/无限维信号的非线性滤波，并研究滤波方程解的各种数值逼近。在这项工作中开发的算法具有应用于加拿大工业和军事的潜力。c)我们将用狄利克雷形式系统地研究具有奇异系数的倒向随机微分方程。所得结果可应用于数学金融的一些问题。

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.