The stochastic nonlinear Schrödinger equation in unbounded domains and manifolds

In this article, we construct a global martingale solution to a general nonlinear Schrödinger 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.