Mixing for acoustic wave motion with boundary random force

TL;DR

This paper investigates the long-term behavior of acoustic wave systems influenced by boundary-driven white noise, establishing conditions for unique invariant measures and convergence of solutions.

Contribution

It introduces an abstract framework linking Markov semigroup regularity to observability and applies it to prove mixing and invariant measure existence for boundary-perturbed acoustic waves.

Findings

Existence of a unique invariant measure for the stochastic wave system

Weak convergence of the system's law to the invariant measure

Connection between semigroup regularity and observability property

Abstract

This paper concerns about the large time behavior of acoustic wave motion driven by a random force acting through the boundary. We begin with an abstract result showing the interconnection between the regularity of Markov semigroup generated by a stochastic evolution equation and the observability property of the corresponding adjoint system. This result is then applied to study the mixing for acoustic wave system with a boundary random perturbation of the white noise type. We shall show there exists a unique invariant measure for the stochastic wave system, and the law of the solution to the system converges to this invariant measure weakly.