Polynomial mixing for the white-forced wave equation on the whole line

TL;DR

This paper proves polynomial mixing and uniqueness of stationary measures for the white-forced stochastic wave equation on the entire real line, overcoming challenges due to lack of smoothing and compactness.

Contribution

It introduces a new criterion for polynomial mixing, a weight Foiaș-Prodi estimate, and weight energy estimates specific to the stochastic wave equation on the whole line.

Findings

Proved polynomial mixing in the dual-Lipschitz metric.

Established uniqueness of stationary measure.

Developed new estimates overcoming lack of smoothing and compactness.

Abstract

Our goal in this paper is to investigate ergodicity of the white-forced wave equation on the whole line. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we prove the uniqueness of stationary measure and polynomial mixing in the dual-Lipschitz metric. The difficulties in our proof are twofold. On the one hand, compared to stochastic parabolic equation, stochastic wave equation is lack of smoothing effect and strongly dissipative mechanism. On the other hand, the whole line leads to the lack of compactness compared to bounded domain. In order to overcome the above difficulties, our proof is based on a new criterion for polynomial mixing established in [20], a new weight type Foia\c{s}-Prodi estimate of wave equation on the whole line and weight energy estimates for stochastic wave equation.