Polynomial mixing of a stochastic wave equation with dissipative damping

TL;DR

This paper investigates the long-term statistical behavior of a semi-linear stochastic wave equation with dissipative damping, showing that the system exhibits polynomial mixing rates under certain conditions.

Contribution

It demonstrates polynomial mixing rates for a class of stochastic wave equations with nonlinear damping, extending known results from reaction-diffusion systems to wave equations.

Findings

The system has at least polynomial mixing rates of any order.

Lyapunov conditions and contraction properties are key to establishing mixing.

The results apply to models with additive Gaussian noise and dissipative damping.

Abstract

We study the long time statistics of a class of semi--linear wave equations modeling the motions of a particle suspended in continuous media while being subjected to random perturbations via an additive Gaussian noise. By comparison with the nonlinear reaction settings, of which the solutions are known to possess geometric ergodicity, we find that, under the impact of nonlinear dissipative damping, the mixing rate is at least polynomial of any order. This relies on a combination of Lyapunov conditions, the contracting property of the Markov transition semigroup as well as the notion of $d$--small sets.