Markovian reduction and exponential mixing in total variation for random dynamical systems

TL;DR

This paper proves that discrete-time random dynamical systems with Markovian noise in finite-dimensional compact spaces converge to a unique invariant measure in total variation, using Markovian reduction and mixing properties.

Contribution

It introduces a novel approach combining Markovian reduction and mixing to establish convergence to a unique measure for such systems.

Findings

Trajectories converge to a unique invariant measure in total variation

The method applies to systems driven by stationary noises

Provides conditions under which exponential mixing occurs

Abstract

The paper deals with the problem of large-time behaviour of trajectories for discrete-time dynamical systems driven by a random noise. Assuming that the phase space is finite-dimensional and compact, and the noise is a Markov process with a transition probability satisfying some regularity hypotheses, we prove that all the trajectories converge to a unique measure in the total variation metric. The proof is based on the Markovian reduction of the system in question and a result on mixing for Markov processes. Then we present an extension of this result to the case of systems driven by stationary noises.