Classical solution to a multidimensional stochastic Burgers equation via forward-backward SDEs

TL;DR

This paper establishes existence, uniqueness, and gradient estimates for a multidimensional stochastic Burgers equation using forward-backward SDEs, and explores its behavior in the vanishing viscosity limit.

Contribution

It introduces a novel approach to solving the stochastic Burgers equation without gradient assumptions, employing a transformation to a random PDE and forward-backward SDEs.

Findings

Proved existence and uniqueness of classical solutions.

Derived a new a priori gradient estimate for quasilinear PDEs.

Analyzed the vanishing viscosity limit of the stochastic Burgers equation.

Abstract

In this paper, we address the problem of existence and uniqueness of a global classical solution to a multidimensional stochastic Burgers equation without gradient-type assumptions on the force or the initial condition. The equation is first transformed to a random PDE, and then solved via the associated forward-backward SDE. Additionally, we obtain a new a priori gradient estimate valid for a large class of second-order quasilinear parabolic PDEs which becomes an important tool in our approach. Also, we study the stochastic Burgers equation in the vanishing viscosity limit.