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

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