M Hairer and NS Pillai
Ergodicity of hypoeliptic SDEs driven by fractional Brownian motion
Abstract: We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 12 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying H¨ormander’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.
Keywords: Ergodicity, Fractional Brownian motion, H¨ormander’s theorem.