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Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

  • Zdzisław Brzeźniak
  • Fabian HornungEmail author
  • Utpal Manna
Article
  • 66 Downloads

Abstract

We construct a martingale solution of the stochastic nonlinear Schrödinger equation (NLS) with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in \(H^1\) on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood–Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of càdlàg  functions and Jakubowski’s generalization of the Skorohod-Theorem to nonmetric spaces.

Keywords

Nonlinear Schrödinger equation Weak martingale solutions Marcus canonical form Lévy noise Littlewood–Paley decomposition 

Mathematics Subject Classification

60H15 35R60 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.The University of YorkHeslington, YorkUK
  2. 2.Institute for AnalysisKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  3. 3.School of MathematicsIndian Institute of Science Education and Research ThiruvananthapuramThiruvananthapuramIndia

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