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

Zdzislaw Brzezniak; Fabian Hornung; Utpal Manna

doi:10.1007/s40072-019-00141-x

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

Zdzislaw Brzezniak, Fabian Hornung, Utpal Manna

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Number of pages	53
Journal	Stochastic Partial Differential Equations: Analysis and Computations
Early online date	16 May 2019
DOIs	https://doi.org/10.1007/s40072-019-00141-x
Publication status	E-pub ahead of print - 16 May 2019

Bibliographical note

© 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.

Keywords

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

Access to Document

10.1007/s40072-019-00141-x

BHM_2019_SNSE-Levy
© 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.
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Cite this

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title = "Weak martingale solutions for the stochastic nonlinear Schrodinger equation driven by pure jump noise",

abstract = "We construct a martingale solution of the stochastic nonlinear Schr{\"o}dinger equation(NLS) with a multiplicative noise of jump type in the Marcus canonical form. Theproblem 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 c{\`a}dl{\`a}g functions and Jakubowski{\textquoteright}s generalization of the Skorohod-Theorem to nonmetric spaces.",

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author = "Zdzislaw Brzezniak and Fabian Hornung and Utpal Manna",

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AU - Hornung, Fabian

AU - Manna, Utpal

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N2 - We construct a martingale solution of the stochastic nonlinear Schrödinger equation(NLS) with a multiplicative noise of jump type in the Marcus canonical form. Theproblem 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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KW - Lévy noise

KW - Littlewood–Paley decomposition

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