Weak martingale solutions for the stochastic nonlinear Schr\"odinger equation driven by pure jump noise

TL;DR

This paper constructs weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by jump noise, using a Faedo-Galerkin approach and advanced probabilistic techniques.

Contribution

It introduces a novel framework for solving stochastic NLS with jump noise in a general setting, including compact manifolds and bounded domains.

Findings

Successfully constructs martingale solutions for stochastic NLS with jump noise.

Employs a Faedo-Galerkin method with Littlewood-Paley operators for uniform estimates.

Utilizes tightness criteria and Skorohod's theorem in nonmetric spaces.

Abstract

We construct a martingale solution of the stochastic nonlinear Schr\"odinger equation with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood-Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of cadlag functions and Jakubowski's generalization of the Skorohod-Theorem to nonmetric spaces.