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We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa-Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in H m (m ∈ N) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-Itô correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Itô formula and a DiPerna-Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

We survey some recent developments in the analysis of the long-time behavior of stochastic solutions of nonlinear conservation laws driven by stochastic forcing. Moreover, we establish the existence and uniqueness of invariant measures for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white noises. We also discuss some further developments, problems, and challenges in this direction.

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