# The stochastic nonlinear Schrödinger equation in unbounded domains and manifolds

Hornung, Fabian

##### Abstract:
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.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2019 Sprache Englisch Identifikator ISSN: 2365-662X KITopen-ID: 1000095974 Verlag KIT, Karlsruhe Umfang 39 S. Serie CRC 1173 ; 2019/11 Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015) Schlagworte nonlinear Schrödinger equation, multiplicative noise, Stratonovich noise, martingale solution, generalized Galerkin approximation, weak compactness method
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