In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on \({\mathbb {R}}^3\) with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.

中文翻译：

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随机非线性薛定ding方程的几乎守恒律

在本文中，我们通过使I方法（几乎守恒定律的方法）适应随机环境，提出了带有加性噪声的随机非线性色散PDE的全球化论点。作为模型示例，我们考虑\（{\ mathbb {R}} ^ 3 \）上的散焦随机三次非线性Schrödinger方程（SNLS），具有加性随机强迫，时间为白色，并且在空间上相关，因此噪声低于能量空间。通过将I方法与Ito引理和停止时间参数相结合，我们为能量空间以下的SNLS构造了实时全局动力学。