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

TL;DR

This work extends the Feynman-Kac formula to stochastic PDEs, establishing existence, uniqueness, and the semigroup property of viscosity solutions for Kolmogorov backward equations, with applications to stochastic Burgers and 2D Navier-Stokes equations.

Contribution

It generalizes the Kolmogorov backward equation framework to SPDEs with non-globally Lipschitz nonlinearities and degenerate diffusion, linking solutions to the underlying SPDEs.

Findings

Proved existence and uniqueness of viscosity solutions for Kolmogorov equations in SPDEs.

Established the semigroup property linking solutions of Kolmogorov equations and SPDEs.

Applied results to stochastic Burgers and 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.