Polynomial mixing for the white-forced hyperviscous Burgers equation on the whole line

Abstract

Our goal in this paper is to investigate ergodicity of the white-forced hyperviscous Burgers equation on the whole line. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we can prove that the dynamics is attractive toward a unique invariant probability measure with polynomial rate of any power. In order to prove this, we further develop coupling method and establish a sufficiently general criterion for polynomial mixing. The proof of polynomial mixing for the white-forced hyperviscous Burgers equation is obtained by the combination of the coupling criterion and the Foiaş-Prodi estimate of hyperviscous Burgers equation on the whole line.