# The stochastic nonlinear Schr\"odinger equation in unbounded domains and manifolds

## TL;DR

This paper establishes the existence of global solutions for a broad class of stochastic nonlinear Schrödinger equations on various unbounded and manifold domains, using advanced probabilistic and spectral methods.

## Contribution

It introduces a novel framework for solving stochastic nonlinear Schrödinger equations on unbounded domains and manifolds, including new approximation and tightness techniques.

## Key findings

- Constructed global martingale solutions for the equation.
- Applicable to diverse unbounded spatial domains and manifolds.
- Handles subcritical nonlinearities with energy space initial data.

## Abstract

In this article, we construct a global martingale solution to a general nonlinear Schr\"{o}dinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like $\mathbb{R}^d$, non-compact Riemannian manifolds, and unbounded domains in $\mathbb{R}^d$ with different boundary conditions. The initial value belongs to the energy space $H^1$ and we treat subcritical focusing and defocusing power nonlinearities. The proof is based on an approximation technique which makes use of spectral theoretic methods and an abstract Littlewood-Paley-decomposition. In the limit procedure, we employ tightness of the approximated solutions and Jakubowski's extension of the Skorohod Theorem to nonmetric spaces.