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arXiv · 1906.08638

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

Fabian Hornung
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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.

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Fabian Hornung. 2019-06-20. The stochastic nonlinear Schr\"odinger equation in unbounded domains and manifolds. https://arxiv.org/abs/1906.08638

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