Rate of convergence to equilibrium for discrete-time stochastic dynamics with memory

TL;DR

This paper investigates the long-term behavior of discrete-time stochastic systems with memory driven by Gaussian noise, establishing conditions for convergence to equilibrium and analyzing how the noise's covariance influences the rate.

Contribution

It proves existence and uniqueness of invariant measures for such systems and provides explicit bounds on convergence rates based on the noise's covariance structure, including fractional dynamics.

Findings

Established conditions for invariant measure existence and uniqueness.

Derived bounds on convergence rates linked to covariance decay.

Applied results to fractional dynamics, including Euler schemes for fractional SDEs.

Abstract

The main objective of the paper is to study the long-time behavior of general discrete dynamics driven by an ergodic stationary Gaussian noise. In our main result, we prove existence and uniqueness of the invariant distribution and exhibit some upper-bounds on the rate of convergence to equilibrium in terms of the asymptotic behavior of the covariance function of the Gaussian noise (or equivalently to its moving average representation). Then, we apply our general results to fractional dynamics (including the Euler Scheme associated to fractional driven Stochastic Differential Equations). Whenthe Hurst parameter H belongs to (0, 1/2) we retrieve, with a slightly more explicit approach due to the discrete-time setting, the rate exhibited by Hairer in a continuous time setting. In this fractional setting, we also emphasize the significant dependence of the rate of convergence to equilibriumon the local behaviour of the covariance function of the Gaussian noise.