Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics

# Project B4

## Kolmogorov operators and SPDE

Principal Investigator(s) Other Investigators

## Summary:

The aim of the project is (a) to develop a theory providing analytic techniques to solve Kolmogorov and Fokker-Planck equations in infinite dimensions and reconstruct from their solutions a solution to the associated stochastic partial differential equations (SPDE), and (b) to solve and analyse the SPDE directly in case of more regular coefficients. Both will be done further developing several approaches which are in case (a) an approach via $L^p$-spaces with respect to an excessive measure of the Kolmogorov operator L and an approach based on a suitably newly formulated maximum principle for L on weighted spaces of weakly continuous functions, and in case (b) both the variational and semigroup (mild solution) approach. In particular, the spectral analysis and geometry of the Kolmogorov operators will be central points of the research. Among the main further issues are: existence and uniqueness of (infinitesimally) invariant measures, spectral properties and functional inequalities for L, large time asymptotics, jump type and other noises, small noise large deviations, finite speed of propagation, stochastic boundary dissipation, applications to SPDE from hydrodynamics and to Kolmogorov operators of particle systems.

## Recent Preprints:

|       |                                                                                                                                    |             |
|-------|------------------------------------------------------------------------------------------------------------------------------------|-------------|
| 13039 | Stochastic nonlinear Schrödinger equations with linear multiplicative noise: the rescaling approach                                | PDF | PS.GZ |
| 13035 | On continuity equations in infinite dimensions with non-gaussian reference measure                                                 | PDF | PS.GZ |
| 13034 | On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures                   | PDF | PS.GZ |
| 13033 | Stochastic Generalized Porous Media Equations with Reflection                                                                      | PDF | PS.GZ |
| 13031 | Random attractor associated with the quasi-geostrophic equation                                                                    | PDF | PS.GZ |
| 13019 | Finite speed of propagation for stochastic porous media equations                                                                  | PDF | PS.GZ |
| 13011 | Exact null internal controllability for the heat equation on unbounded convex domains                                              | PDF | PS.GZ |
| 13010 | On extensions of Sobolev functions on infinite-dimensional spaces                                                                  | PDF | PS.GZ |
| 12144 | On estimates of solutions of Fokker–Planck–Kolmogorov equations with potential terms and non uniformly elliptic diffusion matrices | PDF | PS.GZ |
| 12143 | Nonlinear parabolic equations for probability measures                                                                             | PDF | PS.GZ |
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