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.

 17021 Well-posedness by noise for scalar conservation laws PDF | PS.GZ 16059 Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE PDF | PS.GZ 16058 Stochastic non-isotropic degenerate parabolic-hyperbolic equations PDF | PS.GZ 16057 Regularization by noise for stochastic Hamilton-Jacobi equations PDF | PS.GZ 16041 Representations of Solutions to Fokker-Planck-Kolmogorov Equations with Coefficients of Low Regularities PDF | PS.GZ 16040 A splitting algorithm for stochastic partial differential equations driven by linear multiplicative noise PDF | PS.GZ 16034 Ergodicity for the stochastic quantization problems on the 2D-torus PDF | PS.GZ 16032 Three-dimensional Navier-Stokes equations driven by space-time white noise PDF | PS.GZ 16031 Approximating three-dimensional Navier-Stokes equations driven by space-time white noise PDF | PS.GZ 16030 Piecewise linear approximation for the dynamical $\Phi^4_3$ model PDF | PS.GZ