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

By Martin Hairer, Natesh S. Pillai. 2011

83 citations recorded by the source corpus.

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 2 ( 1 , 1 ). Our contribution in this work is twofold. First, when the driving vector fields satisfy Hormander's c elebrated "Lie bracket condition", we derive explicit quantitative bounds on the i nverse of the Malliavin ma- trix. 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 condi tion, the solutions to rough differential equations driven by fractional Brownian motion with H 2 ( 1 , 1 ) 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".

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5e76bad8268b5cf1cd1fea7bc3cb3316c1defed6

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Citations in the corpus

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in $(1/2,1)$ (2017, background): “In the articles [11, 12, 13, 14] the existence of adapted stationary solutions to dissipative finite-dimensional SDEs driven by fBm and their speed of convergence to the stationary state is studied, by extending Markovian notions as strong Feller property, invariant measure and adaptedness to the no”
Integrability Estimates for Gaussian Rough Differential Equations (2011, background): “Hairer and Pillai [19] have also proved H¨ormander-type theorems for a general class of RDEs; their results are predicated on the assumption that the Jacobian has ﬁnite moments of all order.”
A theory of regularity structures (2013, background): “See [68] for more details in the particular case of rough paths.”
HIGH ORDER PARACONTROLLED CALCULUS (2016, background): “e white noise in M, for d“ 2 or 3, then Rpζq is almost surely p4 ´ dq´-truly rough. A suﬃcient condition for a function for being truly rough is provided by Hairer-Pillai’s notion of θ-rough function [21]; see for instance section 6.4 of FrizHairer’s lecture notes [11]. It may be interesting to note”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, methodology): “This is the content of our second main contribution, which can be seen as a generalisation of the Norris lemma from [23] to a much”
Integrability of (Non-)Linear Rough Differential Equations and Integrals (2011, methodology): “ng such integrability questions was made recently by Cass–Litterer–Lyons [5] (under the standing assumption of bounded, Lipregular vector ﬁelds) driven by suitable Gaussian rough paths; Hairer-Pillai [17] were so able to establish smooth densities of (non-linear) rough diﬀerential equations driven b”