Abstract

 I will speak about the existence of Weak Martingale Solutions for the Stochastic Nonlinear fractional Schrödinger Equations on riemannian manifolds.  These results  are  special cases of general results obtained jointly with Lutz Weis and Fabian Hornung. I will also speak about the existence of invariant measures and stationary solutions to such equations  obtained jointly with Benedetta Ferrario and Margherita Zanella. I will conclude, if time allows talking about the uniqueness results which so far have been proved for Stochastic Nonlinear (non-fractional) Schrödinger Equations and which generalisations to Stochastic Nonlinear fractional Schrödinger Equations would be of some interested. References: [1] Z. Brzezniak, F.  Hornung,  and L.  Weis, Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space.  Probab. Theory Related Fields  174,  no. 3-4, 1273—1338 (2019) [2] Z. Brzezniak, B. Ferrario and M. Zanella, Invariant measures for a stochastic nonlinear and damped  2D Schr\”odinger equation, arxiv [3] Z. Brzezniak, F.  Hornung,  and L.  Weis, Uniqueness of martingale solutions for the stochastic nonlinear Schr\”odinger equation on 3d compact manifolds, to appear