Almost conservation laws for stochastic nonlinear Schr\"odinger equations

TL;DR

This paper extends the $I$-method to stochastic nonlinear Schr"odinger equations, enabling the construction of global solutions below the energy space for SNLS with additive noise.

Contribution

It adapts the $I$-method to stochastic dispersive PDEs, providing a new approach for global well-posedness below energy space in stochastic settings.

Findings

Established global dynamics for stochastic NLS below energy space.

Combined $I$-method with Ito's lemma and stopping times.

Achieved global solutions for SNLS with additive noise.

Abstract

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 Schr\"odinger 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.