Damping‐diffusion‐noise interactions in the stochastic Camassa–Holm equation

In this paper, we investigate the effects of the interaction between time‐inhomogeneous damping, non‐local diffusion and noise on classical solutions to the Camassa–Holm equation incorporating these features. First, a local‐in‐time theory is established, covering the existence, uniqueness and a blow‐up criterion under relatively general conditions for these interacting mechanisms. Subsequently, we identify different conditions on the interactions between damping, diffusion and noise that guarantee the global regularity and long‐time behaviour of classical solutions. Notably, we demonstrate the existence of an evolution system of measures, which generalises the concept of invariant measures to the time‐inhomogeneous setting.