Open Access
July 2013 Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
Martin Hairer, Natesh S. Pillai
Ann. Probab. 41(4): 2544-2598 (July 2013). DOI: 10.1214/12-AOP777


We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2}]$. Our contribution in this work is twofold.

First, when the driving vector fields satisfy Hörmander’s celebrated “Lie bracket condition,” we derive explicit quantitative bounds on the inverse of the Malliavin matrix. En route to this, we provide a novel “deterministic” version of Norris’s lemma for differential equations driven by rough paths. This result, with the added assumption that the linearized equation has moments, will then yield that the transition laws have a smooth density with respect to Lebesgue measure.

Our second main result states that under Hörmander’s condition, the solutions to rough differential equations driven by fractional Brownian motion with $H\in(\frac{1}{3},\frac{1}{2}]$ enjoy a suitable version of the strong Feller property. Under a standard controllability condition, this implies that they admit a unique stationary solution that is physical in the sense that it does not “look into the future.”


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Martin Hairer. Natesh S. Pillai. "Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths." Ann. Probab. 41 (4) 2544 - 2598, July 2013.


Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1288.60068
MathSciNet: MR3112925
Digital Object Identifier: 10.1214/12-AOP777

Primary: 60H07 , 60H10
Secondary: 26A33 , 60G10

Keywords: fractional Brownian motion , Hörmander’s theorem , Hypoellipticity , Rough paths

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 4 • July 2013
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