Decay of the stochastic linear Schr\"odinger equation in $d \geq 3$ with small multiplicative noise

TL;DR

This paper establishes decay estimates and asymptotic behavior for solutions to a stochastic linear Schrödinger equation in dimensions three and higher, with small multiplicative noise that is white in time and colored in space.

Contribution

It extends decay analysis techniques to stochastic Schrödinger equations with small multiplicative noise, adapting methods from deterministic cases.

Findings

Decay estimates for solutions in high dimensions

Asymptotic behavior characterization

Application of bootstrapping argument to stochastic setting

Abstract

We give decay estimates of the solution to the linear Schr\"odinger equation in dimension $d \geq 3$ with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the solution. The proof relies on the bootstrapping argument used by Journ\'e-Soffer-Sogge for decay of deterministic Schr\"odinger operators.