非对称狄氏型的应用和随机泛函微分方程相关问题的研究

The one-to-one relationship between Dirichlet forms and right Markov processes builds a bridge between classical potential theory and stochastic analysis. Through this bridge we can make mutual transformations between some analytic problems and stochastic problems. The research contents of this grant include: 1. By using Dirichlet form theory, we will study boundary value problems associated with non-symmetric second order elliptic operators, obtain the existence, uniqueness and regularity of solutions and give probabilistic representation of solutions. 2. By using semi-Dirichlet form theory, we will study complement value problems associated with non-symmetric second order elliptic integro-differential operators with jumps, obtain the properties of solutions and give probabilistic representation of solutions. 3. We will study perturbation of Dirichlet forms and related boundary value problems. 4. By using non-symmetric Dirichlet form and semi-Dirichlet form theory, we will study large deviation problems, and wish to obtain the large deviation principles for general non-symmetric, not necessarily conservative occupying time with generalized Feynman-Kac functional. 5. We will study martingale problems and transportation inequalities of stochastic functional differential equations with infinite time delay, and aim to obtain the existence and uniqueness of several types of these equations, and transportation inequalities under some particular distance.

狄氏型和右过程之间的一一对应关系在经典位势论与随机分析间架设了一座桥梁，通过这个桥梁可以将一些分析问题与随机分析问题相互转化。本项目的主要研究内容有：1、利用非对称狄氏型理论，研究非对称二阶椭圆算子边值问题，得到解的存在唯一性以及正则性，并给出解的概率表示；2、利用半狄氏型理论，研究具有带跳算子的非对称二阶椭圆积分微分算子补值问题，得到相应方程解的性质并给出解的概率表示；3、研究狄氏型扰动与相关边值问题；4、以非对称狄氏型和半狄氏型为工具来研究大偏差问题，得到一般非对称、不必保守和具有广义Feynman-Kac泛函的占位时的大偏差原理；5、研究具有无穷延迟的随机泛函微分方程的鞅问题和运费不等式，得到不同类型的随机泛函微分方程解的存在唯一性以及在特定距离下无穷时滞随机泛函微分方程的运费不等式。