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

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JournalStochastic Partial Differential Equations: Analysis and Computations
DateAccepted/In press - 28 Apr 2019
DateE-pub ahead of print (current) - 16 May 2019
Number of pages53
Early online date16/05/19
Original languageEnglish

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.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.

    Research areas

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

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