Invariant measures for stochastic partial differential equations and asymptotics

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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.