We establish the existence of global martingale solutions of the stochastic Camassa–Holm equation in $H^1(\mathbb{R})$. The construction of the solution is based on the regularization method and the stochastic compactness method. Furthermore, we use Borel–Cantelli Lemma to prove the global existence of mild solution of the stochastic Camassa–Holm equation with small noise in $L^2(\mathbb{R})$.

中文翻译：

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随机Camassa-Holm方程的整体适定性

我们建立了$ H ^ 1（\ mathbb {R}）$中随机Camassa–Holm方程的全局mar解的存在性。解决方案的构建基于正则化方法和随机紧致度方法。此外，我们使用Borel–Cantelli Lemma证明了在$ L ^ 2（\ mathbb {R}）$中具有小噪声的随机Camassa-Holm方程的温和解的整体存在。