Approach to equilibrium for the stochastic NLS

TL;DR

This paper proves exponential convergence to a Gibbs measure for a stochastic nonlinear Schrödinger equation on a torus, using spectral gap techniques, with results uniform in frequency truncation.

Contribution

It establishes a uniform spectral gap for the Fokker-Planck operator of the SNLS, demonstrating exponential approach to equilibrium in both focusing and defocusing cases.

Findings

Proves exponential convergence to Gibbs measure for SNLS.

Establishes a spectral gap uniform in frequency truncation.

Discusses the limit as truncation goes to infinity.

Abstract

We study the approach to equilibrium, described by a Gibbs measure, for a system on a $d$-dimensional torus evolving according to a stochastic nonlinear Schr\"odinger equation (SNLS) with a high frequency truncation. We prove exponential approach to the truncated Gibbs measure both for the focusing and defocusing cases when the dynamics is constrained via suitable boundary conditions to regions of the Fourier space where the Hamiltonian is convex. Our method is based on establishing a spectral gap for the non self-adjoint Fokker-Planck operator governing the time evolution of the measure, which is {\it uniform} in the frequency truncation $N$. The limit $N\to\infty$ is discussed.