Stochastic Burgers' Equation on the Real Line: Regularity and Moment Estimates

TL;DR

This paper studies the stochastic Burgers' equation with multiplicative white noise on the real line, establishing regularity and moment bounds for its solutions using a Feynman-Kac representation.

Contribution

It introduces a novel solution framework via Feynman-Kac representation and derives regularity and moment estimates for the stochastic Burgers' equation on an unbounded domain.

Findings

Established H"older regularity of solutions.

Derived moment estimates for the solutions.

Connected the solution to a stochastic PDE through Hopf-Cole transformation.

Abstract

In this project we investigate the stochastic Burgers' equation with multiplicative space-time white noise on an unbounded spatial domain. We give a random field solution to this equation by defining a process via a kind of Feynman-Kac representation which solves a stochastic partial differential equation such that its Hopf-Cole transformation solves Burgers' equation. Finally, we obtain H\"older regularity and moment estimates for the solution to Burgers' equation.