Invariant measures for stochastic partial differential equations and asymptotics

在许多物理和生物学问题中，而且在研究社会经济系统的演化时，出现了无限维空间上的某些度量。本课题致力于这些度量的数学研究，特别是研究它们作为相伴随机马尔可夫过程的不变度量的产生。通常，这些不变度量所需的对称性对于应用特别有意义，这使得它们同时在分布空间而不是函数空间上得到支持。相应地，相关随机过程具有奇异系数，难以处理。该项目既涉及数学结构，也涉及对这种措施和相关过程的研究。这涉及到解决马尔可夫半群的生成元的存在唯一性等问题，以及寻找由(伪)微分算子给出的福克-普朗克型方程的解。在无限维Dirichlet形式理论、奇异随机偏(伪)微分方程的研究、渐近分析和遍历理论的前沿，新的方法将被开发出来。

In many problems of physics and biology, but also, e.g. in the study of the evolution of socio-economical systems, certain measures on infinite dimensional spaces arise. This project is dedicated to the mathematical study of these measures and in particular, to the investigation of their arising as invariant measures of associated stochastic Markov processes. Often symmetries required for those invariant measures to be of special interest for applications make them at the same time to be supported on distributional instead of function spaces. Accordingly the associated stochastic processes have singular coefficients and are difficult to handle. The project is concerned both with the mathematical construction and the study of such measures and associated processes. This involves solving problems like existence and uniqueness of generators of Markov semigroups and finding solutions of Fokker-Planck type equations, given by (pseudo-)differential operators. New methods will be developed, at the cutting edge of those coming from the theory of infinite dimensional Dirichlet forms, the study of singular stochastic partial (pseudo-) differential equations, asymptotic analysis and ergodic theory.