Scattering for stochastic nonlinear Schr\"odinger equations

TL;DR

This paper investigates the long-term scattering behavior of solutions to stochastic nonlinear Schrödinger equations with various noise conditions, establishing scattering results in different regimes including energy-critical and high probability scenarios.

Contribution

It provides new scattering results for stochastic nonlinear Schrödinger equations under both finite and infinite quadratic variation noise, covering defocusing and large noise cases.

Findings

Solutions scatter at infinity in the energy and pseudo-conformal spaces for finite quadratic variation noise.

High probability scattering results are established for large, non-conservative noise with infinite quadratic variation.

The results include the energy-critical case and extend to all energy-subcritical exponents.

Abstract

We study the scattering behavior of global solutions to stochastic nonlinear Schr\"odinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is defocusing,we prove that the solutions scatter at infinity in the pseudo-conformal space and in the energy space respectively, including the energy-critical case. Moreover, in the case where the noise is large, non-conservative and has infinite quadratic variation, we show that the solutions scatter at infinity with high probability for all energy-subcritical exponents.