Scattering for Stochastic Nonlinear Schrödinger Equations with additive noise

Abstract

We study the scattering for the energy-subcritical stochastic nonlinear Schrödinger equation (SNLS) with additive noise. In particular, we examine the long-time behavior of solutions associated with the noise $\ddot{I}•(x)g(t,\ddot{I}‰)dB(t,\ddot{I}‰)$ formed by a Schwartz function $\ddot{I}•$, and an adapted process $g(t,\ddot{I}‰)$ 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 $\hat{I}\pounds$ for an initial data in $\hat{I}\pounds$; also in $H^1$ for an initial data in $H^1$.