REGULARITY OF LAWS AND ERGODICITY OF HYPOELLIPTIC SDES DRIVEN BY ROUGH PATHS

• 成果类型: Article
• 署名作者: Hairer, Martin; Pillai, Natesh S.
• 署名单位: University of Warwick; Harvard University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP777
• 发表日期: 2013
• 页码: 2544-2598
• 关键词: differential-equations driven; nonequilibrium statistical-mechanics; malliavin calculus; anharmonic chains

摘要

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 is an element of (1/3, 1/2]. Our contribution in this work is twofold. First, when the driving vector fields satisfy Hormander'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 Hormander's condition, the solutions to rough differential equations driven by fractional Brownian motion with H is an element of (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.