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

Abstract

We prove pathwise uniqueness for solutions of the nonlinear Schrödinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, Gérard and Tzvetkov (N. Burq, P. Gérard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. American Journal of Mathematics, 126 (3):569--605, 2004) to the stochastic setting. The proof is based on deterministic and stochastic Strichartz estimates and the Littlewood-Paley decomposition.