Uniqueness of martingale solutions for the stochastic nonlinear Schrödinger equation on 3d compact manifolds

Brzeźniak, Zdzisław; Hornung, Fabian; Weis, Lutz

doi:10.1007/s40072-022-00238-w

Cited by 3 publications

(1 citation statement)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This approach has been recently used by Crisan et al [CFH19] but it still required the use of the Skorokhod embedding theorem. As it was observed in [BHW22], it would be of interest to see if this approach works for the class of stochastic NLS studied in the present paper.…”

Section: Appendix D Yamada-watanabe Theorem For Stochastic Evolution ...mentioning

confidence: 86%

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

Brzeźniak,

Ferrario,

Zanella

2023

Nonlinearity

Self Cite

View full text Add to dashboard Cite

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension two. We construct a martingale solution using a modified Faedo–Galerkin’s method, following Brzeźniak et al (2019 Probab. Theory Relat. Fields 174 1273–338). Then by means of the Strichartz estimates deduced from Blair et al (2008 Proc. Am. Math. Soc. 136 247–56) but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov–Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler (1999 Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78). This is the first result of this type for stochastic nonlinear Schrödinger equation (NLS) on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well.

show abstract

Section: Appendix D Yamada-watanabe Theorem For Stochastic Evolution ...mentioning

confidence: 86%

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

Brzeźniak,

Ferrario,

Zanella

2023

Nonlinearity

Self Cite

View full text Add to dashboard Cite

show abstract

Stochastic nonlinear Schrödinger equations in the defocusing mass and energy critical cases

Zhang

2023

Ann. Appl. Probab.

View full text Add to dashboard Cite

The stochastic nonlinear Schrödinger equation in unbounded domains and manifolds

Hornung¹

2019

View full text Add to dashboard Cite

Uniqueness of martingale solutions for the stochastic nonlinear Schrödinger equation on 3d compact manifolds

Cited by 3 publications

References 44 publications

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

Stochastic nonlinear Schrödinger equations in the defocusing mass and energy critical cases

The stochastic nonlinear Schrödinger equation in unbounded domains and manifolds

Contact Info

Product

Resources

About