Scattering for Stochastic Nonlinear Schr\"odinger Equations with additive noise

TL;DR

This paper investigates the long-time behavior of solutions to the stochastic nonlinear Schrödinger equation with additive noise, proving almost sure scattering in various functional spaces for initial data with specific regularity.

Contribution

It establishes almost sure scattering results for energy-subcritical SNLS with additive noise in $L^2$, $ ext{pseudo-conformal space } \, ext{and } H^1$, extending understanding of stochastic dispersive equations.

Findings

Proves almost sure scattering in $L^2$ and $\, ext{pseudo-conformal space } \, ext{for initial data in } \, ext{Sigma}$.

Establishes scattering in $H^1$ for initial data in $H^1$.

Analyzes long-time behavior of SNLS with additive noise under decay conditions.

Abstract

We study the scattering for the energy-subcritical stochastic nonlinear Schr\"odinger equation (SNLS) with additive noise. In particular, we examine the long-time behavior of solutions associated with the noise $\phi(x)g(t,\omega)dB(t,\omega)$ formed by a Schwartz function $\phi$, and an adapted process $g(t,\omega)$ satisfying certain decay. Essentially, the aim of the current paper is to prove almost sure scattering in the spaces $L^2$ and the pseudo-conformal space $\Sigma$ for an initial data in $\Sigma$; also in $H^1$ for an initial data in $H^1$.