We consider a class of stochastic evolution equations that include 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 \(s>3/2\). Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.

我们考虑一类随机演化方程，尤其包括随机Camassa–Holm方程。用于在环面的初始值问题，我们首先建立pathwise解的局部存在性和唯一在Sobolev空间\（H 2 -S \）与\（S> 3/2 \）。然后，我们证明足够强大的非线性噪声几乎可以肯定地防止爆炸。为了分析弱噪声的影响，我们考虑了具有非自治前置因子的线性乘法噪声。然后，我们根据初始数据制定精确条件，从而导致全球存在强大的解决方案或导致爆炸。爆裂是由于波浪破裂而发生的。对于具有正概率的爆炸，我们得出这些概率的下限。最后，精确分析了这些解决方案的爆炸率。