The Kolmogorov backward equation for stochastic Burgers equations and for stochastic 2D-Navier-Stokes equations

Abstract

In this book we establish under suitable assumptions the uniqueness and existence of viscosity solutions of Kolmogorov backward equations for stochastic partial differential equations (SPDEs). In addition, we show that this solution is the semigroup of the corresponding SPDE. This generalizes the Feynman-Kac formula to SPDEs and establishes a link between solutions of Kolmogorov equations and solutions of the corresponding SPDEs. In contrast to the literature we only assume that the nonlinear part of the drift is Lipschitz continuous on bounded sets (and not globally Lipschitz continuous) and we allow the diffusion coefficient to be degenerate and non-constant. In the last part of this book we apply our results to stochastic Burgers equations and to stochastic 2-D Navier-Stokes equations.