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

TL;DR

This paper investigates the regularity and ergodic properties of solutions to rough differential equations driven by fractional Brownian motion with Hurst parameter between 1/3 and 1/2, establishing conditions for smooth densities and unique stationary solutions.

Contribution

It provides explicit bounds on the inverse Malliavin matrix and introduces a deterministic Norris's lemma for rough paths, advancing understanding of regularity and ergodicity in rough stochastic systems.

Findings

Transition laws have smooth densities under Hörmander's condition.

Solutions exhibit the strong Feller property.

Existence of a unique stationary solution under controllability.

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 $H\in(\frac{1}{3},\frac{1}{2}]$. Our contribution in this work is twofold. First, when the driving vector fields satisfy H\"{o}rmander'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 H\"{o}rmander'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."