非局部狄氏型和随机偏微分方程若干问题研究

由于狄氏型理论及随机（偏）微分方程在物理学，材料科学、生物学等学科中的应用，关于它们的研究受到了越来越多的重视。本项目拟利用随机分析、泛函分析、调和分析、位势理论和偏微分方程等理论和方法，研究由随机（偏）微分方程确定的连续型、跳跃型马氏过程的热核估计及渐近性质，力图解释相关学科中的科学问题。目标是进一步探索发展狄氏型理论和随机（偏）微分方程的基础理论，并将其应用到具体问题中。拟研究问题如下：(1)跳核带边界行为的狄氏型和马氏过程的位势理论，(2)广义狄氏型与Dirichlet问题，(3)狄氏型与2-维随机欧拉方程，(4)随机偏微分方程解的边界性质、非局部算子相关的马氏过程的分析和概率性质，(5)退化Lévy噪声驱动随机微分方程解的性质，(6)多遍历状态的微分方程在随机扰动下解的遍历性及轨道的概率性质. (7)随机边界的偏微分方程的适定性、解的的正则性和渐进行为.

Due to their applications in physics, material science, biology and other sciences, There is an increasing interest in the study of theory of Dirichlet forms and stochastic (partial) differential equations. This proposal plans to fully use the theory and methods from stochastic analysis, functional analysis, harmonic analysis, potential theory and partial differential equations to study heat kernel estimates and asymptotic behaviors of continuous and jump type Markov processes determined by stochastic (partial) differential equations, try to apply them to important problems in related fields. Our goals are to further develop the theory of Dirichlet forms and stochastic (partial) differential equations, and also apply our theory to concrete problems. This proposal plans to study the following 6 problems: (1) Dirichlet forms with jump kernels approaching zero at the boundary and the potential theory of the corresponding Markov processes, (2) generalized Dirichlet forms and their application to Dirichlet problems, (3) Dirichlet forms and 2 dimensional stochastic Euler equations, (4) boundary properties of solutions of stochastic (partial) differential equations, analytical and probabilistic properties of Markov processes related to non-local operators, (5) properties of solutions of stochastic partial differential equations driven by degenerate Levy noise, (6) ergodicity and sample path properties of stochastic perturbations of differential equations with multiple ergodic states. (7) well-posedness of stochastic boundary value problems of partial differential equations, regularity and asymptotic behavior of solutions.