Global solutions of stochastic nonlinear Schr\"odinger system with quadratic interaction

TL;DR

This paper investigates the global existence of solutions for a stochastic nonlinear Schrödinger system with quadratic interactions, addressing challenges posed by non-conservation of mass and energy due to noise.

Contribution

It extends deterministic results to stochastic systems, establishing global solutions despite the lack of conserved quantities.

Findings

Proved global existence of solutions under stochastic quadratic interactions.

Identified difficulties caused by non-conservation of mass and energy.

Developed new a-priori estimates for stochastic Schrödinger systems.

Abstract

The time-global existence of solutions to a system of stochastic Schr\"odinger equations with multiplicative noise and the quadratic nonlinear terms are discussed in this paper. The same system in the deterministic treatment was studied in [18] where the mass and energy are conserved. In our stochastic situation, those are not conserved and which causes several difficulties in the arguments of composing a-priori estimate.