On the unique ergodicity for a class of 2 dimensional stochastic wave equations

TL;DR

This paper proves the existence and uniqueness of an invariant measure for a 2D stochastic wave equation with cubic nonlinearity and regular noise, extending previous results to cases where the invariant measure is not explicitly known.

Contribution

It establishes the first proof of unique ergodicity for this class of stochastic wave equations with non-explicit invariant measures.

Findings

Existence of an invariant measure for the stochastic wave equation.

Uniqueness of the invariant measure (ergodicity).

Extension of previous results to more general noise settings.

Abstract

We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the $2$-dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author to a situation where the invariant measure is not explicitly known.