Uniqueness of martingale solutions for the stochastic nonlinear Schr\"odinger equation on 3d compact manifolds

TL;DR

This paper establishes the pathwise uniqueness of solutions to the stochastic nonlinear Schrödinger equation on 3D compact manifolds, extending deterministic results to the stochastic context using advanced harmonic analysis techniques.

Contribution

It generalizes existing deterministic uniqueness results for nonlinear Schrödinger equations to the stochastic setting on 3D manifolds.

Findings

Proves pathwise uniqueness for stochastic NLS on 3D manifolds.

Extends deterministic Strichartz estimates to stochastic equations.

Utilizes Littlewood-Paley decomposition in the proof.

Abstract

We prove pathwise uniqueness for solutions of the nonlinear Schr\"{o}dinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, G\'erard and Tzvetkov (N. Burq, P. G\'erard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schr\"{o}dinger 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.