On blow up NLS with a multiplicative noise

TL;DR

This paper investigates the effect of multiplicative noise on the blow-up behavior of nonlinear Schrödinger equations, showing that the probability of rapid blow-up due to noise is very small.

Contribution

It provides a large deviation upper bound demonstrating that noise-induced rapid blow-up is unlikely, refining previous understanding of noise effects on NLS.

Findings

Probability of rapid blow-up due to noise is very small

Established a large deviation type upper bound

Contrasts with earlier results showing noise accelerates blow-up

Abstract

It is of significant interest to understand whether a noise will speed up or prevent blow up. Under certain nondegenerate conditions, \cite{dD2005Blowup} proved a multiplicative noise will speed up blow up of NLS, in the sense that, blow up can happen in any short time with positive probability. We prove that such probability is indeed quite small, and provide a large deviation type upper bound.