Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations

TL;DR

This paper introduces the concepts of random quasi-periodic paths and measures for stochastic differential equations, providing conditions for their existence, uniqueness, and ergodic properties, and linking them to the Fokker-Planck equation.

Contribution

It defines new notions of quasi-periodic structures in stochastic systems and establishes conditions for their existence, uniqueness, and ergodicity, extending the understanding of long-term behavior.

Findings

Existence and uniqueness of random quasi-periodic paths established.

Conditions for the density of quasi-periodic measures and their relation to Fokker-Planck equation provided.

Proved the ergodicity and uniqueness of the invariant measure for the system.

Abstract

In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the invariant measure is unique, and thus ergodic.