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arXiv · 1910.14558

Almost conservation laws for stochastic nonlinear Schr\"odinger equations

Kelvin Cheung·Guopeng Li·Tadahiro Oh
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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.

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Kelvin Cheung, Guopeng Li, Tadahiro Oh. 2019-10-31. Almost conservation laws for stochastic nonlinear Schr\"odinger equations. https://arxiv.org/abs/1910.14558

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