Quantitative Bounds on the Rate of Approach to Equilibrium for some One-Dimensional Stochastic Non-Linear Schr\"odinger Equations

TL;DR

This paper derives quantitative bounds on how quickly certain one-dimensional stochastic nonlinear Schrödinger equations approach equilibrium, providing insights into their long-term behavior with applications to both focusing and defocusing cases.

Contribution

It introduces new methods to quantify the rate of convergence to equilibrium for infinite-dimensional stochastic NLS equations, extending previous results to more complex nonlinearities.

Findings

Established explicit bounds on convergence rates to equilibrium.

Applied analysis to both focusing and defocusing nonlinearities.

Enhanced understanding of stochastic NLS dynamics with diffusive forcing.

Abstract

We establish quantitative bounds on the rate of approach to equilibrium for a system with infinitely many degrees of freedom evolving according to a one-dimensional focusing nonlinear Schr\"odinger equation with diffusive forcing. Equilibrium is described by a generalized grand canonical ensemble. Our analysis also applies to the easier case of defocusing nonlinearities