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

Abstract

Our goal in this paper is to investigate the ergodicity of the white-forced wave equation on the whole line. Assuming that sufficiently many directions of the phase space are stochastically forced, we prove the uniqueness of the stationary measure and polynomial mixing in the dual-Lipschitz metric. The difficulties in our proof are twofold. On the one hand, in contrast to stochastic parabolic equations, the stochastic wave equation lacks both a smoothing effect and a strongly dissipative mechanism. On the other hand, the unboundedness of the whole line leads to a loss of compactness compared with the bounded domain case. To overcome these obstacles, our proof relies on four key ingredients: a new criterion for polynomial mixing established in [21], a novel weighted Foiaş-Prodi estimate for the wave equation on the whole line, weighted energy estimates for the stochastic wave equation, and the irreducibility of the stochastic wave equation.