The Kolmogorov operator associated to a Burgers SPDE in spaces of continuous functions

TL;DR

This paper investigates the Kolmogorov operator linked to a stochastic Burgers equation with white noise perturbation, establishing its properties in a weighted continuous function space and applying these results to solve the Fokker-Planck equation.

Contribution

It characterizes the infinitesimal generator of the transition semigroup for the stochastic Burgers equation in a weighted continuous space and connects it to the Kolmogorov operator.

Findings

The generator is the closure of the Kolmogorov operator in a suitable topology.

The results enable solving the associated Fokker-Planck equation.

Provides a framework for analyzing stochastic PDEs with white noise perturbations.

Abstract

We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.