Stochastic nonlinear Schr\"odinger equations: no blow-up in the non-conservative case

TL;DR

This paper demonstrates that adding large multiplicative Gaussian noise to stochastic nonlinear Schrödinger equations can prevent blow-up solutions with high probability, especially in non-conservative, focusing mass-critical cases, contrasting with conservative scenarios.

Contribution

It proves that noise can prevent blow-up in non-conservative stochastic Schrödinger equations, extending previous work on well-posedness and exploring noise effects on solution behavior.

Findings

High probability of blow-up prevention with large noise

Prevention of blow-up on finite and infinite time intervals

Distinct noise effects compared to conservative case

Abstract

This paper is devoted to the study of noise effects on blow-up solutions to stochastic nonlinear Schr\"odinger equations. It is a continuation of our recent work \cite{BRZ14}, where the (local) well-posedness is established in $H^1$, also in the non-conservative critical case. Here we prove that in the non-conservative focusing mass-(super)critical case, by adding a large multiplicative Gaussian noise, with high probability one can prevent the blow-up on any given bounded time interval $[0,T]$, $0<T<\9$. Moreover, in the case of spatially independent noise, the explosion even can be prevented with high probability on the whole time interval $[0,\9)$. The noise effects obtained here are completely different from those in the conservative case studied in \cite{BD03}.