The energy-critical stochastic nonlinear Schr\"odinger equation: well-posedness and blow-up

TL;DR

This paper studies the energy-critical stochastic nonlinear Schrödinger equation, establishing local well-posedness and blow-up criteria under random perturbations, extending deterministic results to stochastic scenarios.

Contribution

It introduces new well-posedness results and blow-up criteria for stochastic energy-critical NLS with additive or multiplicative noise, extending classical deterministic findings.

Findings

Proved local well-posedness for initial data in Sobolev spaces.

Derived blow-up criteria with positive probability under small noise.

Extended deterministic blow-up results to stochastic energy-critical NLS.

Abstract

We investigate the focusing and defocusing energy-critical stochastic nonlinear Schr\"odinger equation, subject to random perturbations in the form of either additive or multiplicative (Stratonovich) noise. We establish local well-posedness for random or deterministic initial data $u_0$ in $\dot{H}^1(\mathbb{R}^n)$ or $H^1(\mathbb{R}^n)$, depending on the noise type. In the focusing case we provide quantitative estimates regarding the existence time and probability. Moreover, we derive blow-up criteria for solutions with positive energy in both cases of noise, provided that the noise intensity is sufficiently small, showing that blow-up occurs before a certain given positive time with positive probability, thus, extending deterministic results of Kenig-Merle [24] for the energy-critical NLS equation to the stochastic setting.