Abstract
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.
Highlights
The article is concerned with the following nonlinear stochastic Schrödinger equation du(t) = (−i Au(t) − iF(u(t))) dt − iBu(t) ◦ dW (t), t > 0, (1.1)
This section is devoted to the compactness results which will be used to get a martingale solution of (1.1) by the Faedo–Galerkin method
We introduce the Galerkin approximation, which will be used for the proof of the existence of a solution to (1.1)
Summary
The main aim of this study is twofold It proposes to construct a martingale solution of problem (1.1) by a stochastic version of a compactness method. The present paper is motivated by the construction of a global solution of the cubic equation on compact 3d-manifolds M generalizing the existence part, see Theorem 3 of Burq et al [4], to the stochastic setting. We remark that in the mean time, a similar construction has been used in [29] to construct a solution of a stochastic nonlinear Maxwell equation by estimates in Lq for some q > 2 This indicates that our method has potential to increase the field of application of the classical Faedo–.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
