Invariant measures for stochastic partial differential equations and asymptotics

随机偏微分方程和渐近方程的不变测度

• 批准号: 407748683
• 负责人: Professor Dr. Sergio Albeverio
• 金额: --
• 依托单位: Institut für Angewandte Mathematik (IAM)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/407748683?language=en
• 关键词: Invariant; measures; stochastic; partial; differential

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.

在物理学和生物学的许多问题中，以及在研究社会经济系统的演化时，会出现无限维空间上的某些测度。该项目致力于这些措施的数学研究，特别是，调查其产生的相关随机马尔可夫过程的不变措施。通常，对称性要求这些不变的措施是特别感兴趣的应用程序，使他们在同一时间得到支持的分布，而不是功能空间。相应地，相关的随机过程具有奇异系数并且难以处理。该项目既涉及数学结构，也涉及对这些措施和相关过程的研究。这涉及到解决问题，如存在性和唯一性的发电机的马尔可夫半群和寻找解决方案的福克-普朗克型方程，由（伪）微分算子。新的方法将被开发，在那些来自无穷维Dirichlet形式的理论，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。