Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation

TL;DR

This paper proves that a specific Gaussian measure related to a stochastic nonlinear wave equation on a 2D torus remains quasi-invariant under the nonlinear flow, extending understanding of measure behavior in stochastic PDEs.

Contribution

It establishes the quasi-invariance of the Gaussian measure for the 2D stochastic cubic nonlinear wave equation, a novel result in the analysis of stochastic PDEs.

Findings

Gaussian measure is quasi-invariant under the nonlinear flow

Unique invariant measure identified for the linear equation

Extension of measure invariance concepts to stochastic nonlinear wave equations

Abstract

We consider the stochastic damped nonlinear wave equation $\partial_t^{2}u+\partial_t u+u-\Delta u +u^{3} = \sqrt{2} {\langle{\nabla}\rangle^{-s}} \xi$ on the two-dimensional torus $\mathbb T^2$, where $\xi$ denotes a space-time white noise and $s>0$. We show that the measure $\vec{\mu}_s$ corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.