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We consider a class of stochastic evolution equations that includes in particular the stochastic Camassa--Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces $H^s$ with $s>3/2$. For non-autonomous linear multiplicative noise, we formulate then conditions that lead to global existence of strong solutions and wave-breaking phenomena. Wave-breaking solutions appear in fact with positive probability, and lower bounds for these probabilities are derived. Finally, the blow-up rate of these solutions is precisely analyzed.

On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena