The Annals of Probability

Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths

Martin Hairer and Natesh S. Pillai

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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.”

Article information

Ann. Probab. Volume 41, Number 4 (2013), 2544-2598.

First available in Project Euclid: 3 July 2013

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Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60G10: Stationary processes 26A33: Fractional derivatives and integrals

Hörmander’s theorem hypoellipticity fractional Brownian motion rough paths


Hairer, Martin; Pillai, Natesh S. Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Ann. Probab. 41 (2013), no. 4, 2544--2598. doi:10.1214/12-AOP777.

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