The stochastic nonlinear Schrödinger equation in unbounded domains and manifolds

2019/06/20 by Fabian Hornung, Hornung, Fabian
Mathematics · Economics, Econometrics and Finance · Computer Science · #Advanced Mathematical Physics Problems #Stochastic processes and financial applications #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1906.08638

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 ℝ^d, non-compact Riemannian manifolds, and unbounded domains in ℝ^d with different boundary conditions. The initial value belongs to the energy space H¹ 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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