Invariant Measure of the Camassa-Holm Equation with Linear Multiplicative Noise

Abstract

In this paper, we prove that the solution map of Camassa-Holm equation with linear multiplicative noise $$ \left\{ \begin{array}{l} {\rm d}u+(u\partial_xu+\partial_xP[u])\,{\rm d}t=βu\,{\rm d}W, u(0,x)=u_0(x), P[u]=(1-\partial_x^2)^{-1}\left(u^2+\frac 1 2(\partial_x u)^2\right) \end{array} \right. $$ depends almost surely continuously on the deterministic initial data in $H^s$ for $s>3/2$. Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise.