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The stochastic nonlinear Schrödinger equation in unbounded domains and manifolds

Hornung, Fabian

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.

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DOI: 10.5445/IR/1000095974
Veröffentlicht am 25.06.2019
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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