Global existence of dissipative solutions to the Camassa-Holm equation with transport noise

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa-Holm equation perturbed by a convective, position -dependent, noise term. We establish the first global -in -time existence result for dissipative weak martingale solutions to this SPDE, with general finiteenergy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi -Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second -order Stratonovich-Ito correction term. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).