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 $$\{\begin{array}{l}\mathrm{d}u+(u{\partial }_{x}u+{\partial }_{x}P[u])\mathrm{d}t=\beta u\mathrm{d}W,u(0,x)=u_0(x),P[u]=(1-{\partial }_x^2{)}^{-1}\left(u^2+\frac{1}{2}({\partial }_{x}u{)}^2\right)\end{array}$$ 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.