Ergodic theory for SDEs with extrinsic memory

Abstract

We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob--Khas'minskii theorem. The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a nondegeneracy condition on the noise, such equations admit a unique adapted stationary solution.

Published in at http://dx.doi.org/10.1214/009117906000001141 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Formulae for the derivatives of heat semigroups

Cited by in corpus (19)

• Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
• A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion
• Nonparametric Estimation in Fractional SDE
• Fractional term structure models: No-arbitrage and consistency
• Slow-Fast Systems with Fractional Environment and Dynamics
• Approximation of stationary solutions of Gaussian driven Stochastic Differential Equations
• A general drift estimation procedure for stochastic differential equations with additive fractional noise
• Ergodic properties of a class of non-Markovian processes
• Non-Markovian Reduced Systems for Stochastic Partial Differential Equations: The Additive Noise Case
• Integration by parts formula and applications for SDE driven by fractional Brownian motion
• Asymptotic stability of controlled differential equations. Part II: rough integrals
• On a Calculable Skorokhod's Integral Based Projection Estimator of the Drift Function in Fractional SDE
• Stochastic evolution equations driven by Liouville fractional Brownian motion
• Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion
• Random dynamical systems, rough paths and rough flows
• Classification of random circle homeomorphisms up to topological conjugacy
• Concentration estimates for SPDEs driven by fractional Brownian motion
• Approximation of stationary solutions to SDEs driven by multiplicative fractional noise
• Fractional Brownian flows