无穷维随机微分方程

项目名称： 无穷维随机微分方程

项目编号： No.11301026

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 朱蓉禅

作者单位： 北京理工大学

项目金额： 20万元

中文摘要： 无穷维空间上的随机微分方程，是一个重要的研究课题, 是目前国际上概率论领域的一个热门方向。最近几年由于非局部算子的广泛出现， 因此有很多工作研究带有非局部算子的确定性方程。 考虑到实际中的不确定性以及观测的误差，研究在随机扰动下带有非局部算子的偏微分方程是很重要的研究课题。项目申请人希望开展无穷维随机微分方程的研究尤其是带有非局部算子的随机偏微分方程. 申请人准备通过"强方法"即是对无穷维随机微分方程本身来研究解的各种性质,包括解的存在唯一性,遍历性以及随机动力系统的相关性质。这要求我们改进优化随机偏微分方程中的技巧。另一方面，申请人希望通过研究与随机微分方程对应的无穷维Kolmogorov方程以及相应的Fokker-Planck方程得到当噪声是时空白噪声或漂移项比较奇异时随机微分方程解的存在唯一性。这要求发展对无穷多个变量的函数和测度的偏微分方程的理论来应用到更多现实中的例子。

中文关键词： 随机偏微分方程；遍历性；随机吸引子；Fokker-Planck方程；狄氏型

英文摘要： A large number of models were found that could be described by partial differential equations with random parameters, such as the coefficients or the forcing term. As a result, the study of SDE in infinite dimensional space has begun to attract a lot of attention of many researchers. Recently, since the non-local operators are widely used, there are many researchers studying the partial differential equations with non-local operators. Taking into account of the uncertainties in the actual and the observation error, to study the stochastic partial differential equations with non-local operators is a very important research topic. The project applicant wishes to carry out the study of infinite dimensional stochastic differential equations, especially stochastic partial differential equations with non-local operators. Applicant wants to apply the so-called "strong approach", i.e. through the underlying infinite dimensional stochastic differential equation to study the various properties of the solutions, including existence and uniqueness of solutions, ergodicity, stochastic dynamical systems and related properties. This requires us to improve and refine techniques from the theory of infinite dimensional ordinary differential equations. On the other hand, the applicant wants to use the "weak approach", i.e. th

英文关键词： stochastic partial differential equation；ergodicity；random attractor；Fokker-Planck equation；Dirichlet form