We study the effect of noise on the stochastic modified Camassa–Holm equation. We first derive the stochastic modified Camassa–Holm equation by the stochastic variational principle. Then, we prove the well-posedness of the stochastic modified Camassa–Holm equation by the iterative process. Under the condition of the small noise intensity, we can also get the regularization of the solution from the parabolic term.

中文翻译：

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噪声对随机修正 Camassa-Holm 方程的影响

我们研究了噪声对随机修正 Camassa-Holm 方程的影响。我们首先通过随机变分原理推导出随机修正的 Camassa-Holm 方程。然后，我们通过迭代过程证明了随机修正的 Camassa-Holm 方程的适定性。在噪声强度较小的情况下，我们也可以从抛物线项中得到解的正则化。