The Gaussian Structure of the Singular Stochastic Burgers Equation

TL;DR

This paper investigates the stochastic Burgers equation with rough noise, showing that at fixed times, its law is absolutely continuous with respect to the stochastic heat equation, revealing a Gaussian structure and ellipticity in an infinite-dimensional setting.

Contribution

It introduces a novel approach to analyze the stochastic Burgers equation with rough noise by decomposing solutions and extending Girsanov's theorem, providing new insights into its Gaussian structure.

Findings

Law of the process at fixed time is absolutely continuous w.r.t. stochastic heat equation.

Decomposition of solutions into auxiliary processes with Gaussian law.

Insights into the structure of solutions near KPZ regularity.

Abstract

We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time $t$, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed as we are beyond the regime of classical solutions for much of the paper.