Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

TY - JOUR

T1 - Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

AU - Brzezniak, Zdzislaw

AU - Hornung, Fabian

AU - Weis, Lutz

N1 - © The Author(s) 2018

PY - 2019/8/1

Y1 - 2019/8/1

N2 - We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing Stochastic NLSE in H 1 on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo–Galerkin-method based on the Littlewood–Paley-decomposition. For the 2d manifolds with bounded geometry, we use the Strichartz estimates to show the pathwise uniqueness of solutions.

AB - We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing Stochastic NLSE in H 1 on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo–Galerkin-method based on the Littlewood–Paley-decomposition. For the 2d manifolds with bounded geometry, we use the Strichartz estimates to show the pathwise uniqueness of solutions.

KW - Compactness method

KW - Galerkin approximation

KW - Multiplicative noise

KW - Nonlinear Schrödinger equation

KW - Pathwise uniqueness

UR - http://www.scopus.com/inward/record.url?scp=85055968344&partnerID=8YFLogxK

U2 - 10.1007/s00440-018-0882-5

DO - 10.1007/s00440-018-0882-5

M3 - Article

SN - 1432-2064

VL - 174

SP - 1273

EP - 1338

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -