Exponential mixing for finite-dimensional approximations of the Schr\"odinger equation with multiplicative noise

TL;DR

This paper proves that finite-dimensional approximations of the Schrödinger equation with multiplicative noise exhibit exponential mixing, ensuring rapid convergence to a unique stationary distribution on the unit sphere.

Contribution

It establishes ergodicity and exponential mixing for these approximations under general noise distribution assumptions, a novel result in this context.

Findings

Unique stationary measure exists on the unit sphere.

System converges exponentially fast to the stationary measure.

Stationary measure is absolutely continuous with respect to volume.

Abstract

We study the ergodicity of finite-dimensional approximations of the Schr\"odinger equation. The system is driven by a multiplicative scalar noise. Under general assumptions over the distribution of the noise, we show that the system has a unique stationary measure $\mu$ on the unit sphere $S$ in $\C^n$, and $\mu$ is absolutely continuous with respect to the Riemannian volume on $S$. Moreover, for any initial condition in $S$, the solution converges exponentially fast to the measure $\mu$ in the variational norm.