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Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
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Hairer, Martin and Pillai, Natesh S. (2013) Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. The Annals of Probability, Volume 41 (Number 4). pp. 2544-2598. doi:10.1214/12-AOP777 ISSN 0091-1798.
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Official URL: http://dx.doi.org/10.1214/12-AOP777
Abstract
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 = (1/3, 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 = (1/3, 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.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | The Annals of Probability | ||||
Publisher: | Institute of Mathematical Statistics | ||||
ISSN: | 0091-1798 | ||||
Official Date: | 2013 | ||||
Dates: |
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Volume: | Volume 41 | ||||
Number: | Number 4 | ||||
Page Range: | pp. 2544-2598 | ||||
DOI: | 10.1214/12-AOP777 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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