Regularization by noise for the energy- and mass-critical nonlinear Schr\"odinger equations

TL;DR

This paper demonstrates that adding strong non-conservative noise to energy- and mass-critical nonlinear Schrödinger equations ensures solutions exist globally and scatter with high probability, revealing a regularization by noise effect.

Contribution

It establishes a novel regularization by noise phenomenon for critical nonlinear Schrödinger equations, using rescaling and decay properties of geometric Brownian motions.

Findings

Probability of global existence and scattering approaches one as noise strength increases.

Introduces a new approach combining rescaling transform and Brownian motion decay analysis.

Shows noise can regularize solutions in critical nonlinear Schrödinger equations.

Abstract

In this article we prove a regularization by noise phenomenon for the energy-critical and mass-critical nonlinear Schr\"odinger equations. We show that for any deterministic data, the probability that the corresponding solution exists globally and scatters goes to one as the strength of the non-conservative noise goes to infinity. The proof relies on the rescaling transform and a new observation on the rapid uniform decay of geometric Brownian motions after short time.