Weak martingale solutions for the stochastic nonlinear Schr "odinger\n equation driven by pure jump noise

2018/09/26 by Zdzisław Brzeźniak, Brzeźniak, Zdzisław, Fabian Hornung +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1809.10013

openalex publication_date 2018/09/26 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

We construct a martingale solution of the stochastic nonlinear Schr "odinger\nequation with a multiplicative noise of jump type in the Marcus canonical form.\nThe problem is formulated in a general framework that covers the subcritical\nfocusing and defocusing stochastic NLS in H¹ on compact manifolds and on\nbounded domains with various boundary conditions. The proof is based on a\nvariant of the Faedo-Galerkin method. In the formulation of the approximated\nequations, finite dimensional operators derived from the Littlewood-Paley\ndecomposition complement the classical orthogonal projections to guarantee\nuniform estimates. Further ingredients of the construction are tightness\ncriteria in certain spaces of cadlag functions and Jakubowski's generalization\nof the Skorohod-Theorem to nonmetric spaces.\n

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