Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

Abstract

We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander's condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that they admit a unique stationary solution that is physical in the sense that it does not "look into the future". The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.

To appear in the Annals of IHP

References in corpus (4)

• Formulae for the derivatives of heat semigroups
• Ergodic theory for SDEs with extrinsic memory
• Ergodic properties of a class of non-Markovian processes
• A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs