Exponential Decay of $L^2$-Solutions to Stochastic Nonlinear Schr\"odinger Equations Driven by Continuous Martingales

TL;DR

This paper proves that solutions to stochastic nonlinear Schrödinger equations driven by continuous martingales decay exponentially in the $L^2$ norm, extending known results from Brownian motion to more general martingale noise.

Contribution

It introduces a rescaling method transforming the stochastic equation into a random Schrödinger equation, revealing the stabilizing effect of the martingale's density on solution decay.

Findings

Solutions exist globally in time.

$L^2$-norm decays exponentially with a positive rate.

Decay rate is determined by the martingale's quadratic variation density.

Abstract

We investigate the global well-posedness and asymptotic behavior of $L^2$-solutions to stochastic nonlinear Schr\"odinger equations with multiplicative noise driven by continuous square integrable martingales with density. Our approach relies on a rescaling transformation that converts the stochastic system into a random nonlinear Schr\"odinger equation with a potential acting as a damping term. Unlike the standard Brownian motion case, this induced potential plays a critical role in the dynamics. We establish the global existence of solutions and prove the pathwise exponential decay of the $L^2$-norm. Crucially, the strict positivity of the decay rate is intrinsically induced by the density of the martingale\rq{}s quadratic variation. This result generalizes the stabilization known for standard Brownian motion, thereby characterizing the stabilizing effect of the martingale noise.