In this paper, the stochastic Camassa–Holm equation with pure jump noise in the Marcus form is considered. Firstly, the equation is derived by the stochastic variation method. Then, wave-breaking is shown to occur in expectation. Furthermore, the global existence is obtained under suitable initial conditions. Finally, a moderate deviation principle is proved by the regularized equation, the weak convergence method and an exponential equivalence of the probability measures.

中文翻译：

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具有纯跳跃噪声的随机Camassa–Holm方程的破波和中等偏差

在本文中，考虑了具有纯跳跃噪声的Marcus形式的随机Camassa-Holm方程。首先，通过随机变分法推导该方程。然后，表明发生了预期的波浪破碎。此外，在适当的初始条件下获得了整体存在。最后，通过正则化方程，弱收敛方法和概率测度的指数等价证明了中等偏差原理。