Abstract In this paper, we consider the periodic Cauchy problem for a stochastically perturbed nonlinear dispersive partial differential equation with cubic nonlinearity, which involves the integrable Novikov equation arising from the shallow water wave theory as a special case. We first establish the existence and uniqueness of local pathwise solutions in Sobolev spaces H s ( T ) ( s > 3 2 ) with nonlinear multiplicative noise, where the key ingredients are the stochastic compactness method, the Skorokhod representation theorem and the Gyongy–Krylov characterization of convergence in probability. In the case of linear multiplicative noise, we investigate the conditions which lead to the blow-up phenomena and global existence of pathwise solution. Finally, we show that the linear multiplicative noise has a dissipative effect on the periodic peakon solutions to the associated deterministic Novikov equation.

中文翻译：

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随机扰动非线性色散偏微分方程的局部和全局路径解

摘要 在本文中，我们考虑了一个具有三次非线性的随机摄动非线性色散偏微分方程的周期柯西问题，它涉及到浅水波理论产生的可积Novikov方程作为一个特例。我们首先在具有非线性乘法噪声的 Sobolev 空间 H s ( T ) ( s > 3 2 ) 中建立局部路径解的存在性和唯一性，其中关键成分是随机紧致方法、Skorokhod 表示定理和 Gyongy-Krylov 表征概率收敛。在线性乘法噪声的情况下，我们研究导致爆炸现象和路径解全局存在的条件。最后，