Martingale solutions for the stochastic nonlinear Schr\"odinger equation in the energy space

TL;DR

This paper develops a framework for solving stochastic nonlinear Schrödinger equations with multiplicative noise, establishing existence and uniqueness of solutions in energy spaces on compact manifolds and bounded domains.

Contribution

It introduces a martingale solution construction using a modified Faedo-Galerkin method and proves pathwise uniqueness in 2D manifolds with bounded geometry.

Findings

Existence of martingale solutions for stochastic NLS in energy space.

Pathwise uniqueness established for 2D manifolds with bounded geometry.

Applicable to subcritical focusing and defocusing cases on compact manifolds.

Abstract

We consider a stochastic nonlinear Schr\"odinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo-Galerkin-method based on the Littlewood-Paley-decomposition. For 2d manifolds with bounded geometry, we use Strichartz estimates to show pathwise uniqueness.