Global Existence of Weak Martingale Solutions to the Camassa-Holm Equation with Linear Multiplicative Noise

In this paper, we consider the global existence and properties of $H^1$ martingale solution to the Camassa-Holm equation with linear multiplicative noise under periodic boundary conditions.

The solution is obtained as limit of regular viscous approximate solutions to parabolic SPDEs, which are constructed using the Galerkin approximations ans the stochastic compactness method.

The proof of convergence to a solution argues via tightness of the laws of the viscous approximations and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces.

In particular, by means of the Girsanov-type transform for regular viscous approximations and the convergence of Skorokhod-Jakubowski representations, we are able to establish the one-sided supernorm estimate and space-time higher regularity of the first-order spatial derivative, and large-time behavior of the weak martingale solution in the stochastic framework.