Existence of invariant measures for the stochastic damped Schr\"odinger equation

TL;DR

This paper proves the existence of invariant and ergodic measures for the stochastic damped Schrödinger equation, demonstrating long-term stability and statistical regularity of solutions in a mathematical physics context.

Contribution

It establishes the existence of invariant and ergodic measures for the stochastic damped Schrödinger equation, advancing understanding of its long-term behavior.

Findings

Existence of an invariant measure for the stochastic Schrödinger equation

Asymptotic compactness of solutions

Existence of an ergodic measure

Abstract

In this paper, we address the long time behaviour of solutions of the stochastic Schrodinger equation in $\mathbb{R}^d$. We prove the existence of an invariant measure and establish asymptotic compactness of solutions, implying in particular the existence of an ergodic measure.