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Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

By Martin Hairer and N. S. Pillai


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 Hormander's condition. We show that such systems enjoy a suitable version of the strong Feller property and we conclude that under a standard controllability condition 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

Publisher: Institute of Mathematical Statistics
Year: 2011
DOI identifier: 10.1214/10-aihp377
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