Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

Martin Hairer; Natesh S. Pillai

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arXiv · 0909.4505

Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

Martin Hairer·Natesh S. Pillai

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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.

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Martin Hairer, Natesh S. Pillai. 2010-05-13. Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. https://arxiv.org/abs/0909.4505

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