The aim of the project is to fully develop an analytic approach to the solution of stochastic partial differential equations (SPDE) via the associated Kolmogorov equations (i.e. infinite dimensional generalized heat equations). The corresponding partial differential operators, called Kolmogorov operators, act in infinitely many variables. Their analytic treatment is still at the beginning and several approaches will be implemented during the course of this project. In particular, their spectral analysis and geometry will be central points of the research. Among the important applications, which will constantly provide orientation for the planned work, will be stochastically perturbed partial differential equations (PDE) as e.g. the generalized Burgers, the Navier-Stokes or porous media equations. Specific topics to be investigated also include jump type noises as stochastic perturbations, integrations by parts formulae with boundary terms for invariant measures, SPDE over unbounded domains, non-linear Kolmogorov equations in infinitely many variables, convergence of their spectral structures and non-linear PDE in infinite dimensions, non-linear infinite dimensional state spaces and their geometry/topology.
Stochastic methods are nowadays standardly used in all Natural Sciences as well as as Economics and Engineering. This project will concentrate on the theory and the applications of a branch of Stochastics, namely Stochastic Partial Differential Equations (SPDE), which has undergone an enormous development in recent years. It has become a major research direction in Probability Theory as far as its conceptions and general theory is concerned, and at the same time it is being used more and more in applications in many other disciplines. These include Biology, Chemistry, Physics, in particular Hydrodynamics and Climatology, but also Medical Sciences, Economics, and Engineering. The aims of this project is to contribute to both the mathematical theory of SPDE and to its applications to fundamental problems in the sciences mentioned above.
