Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

TL;DR

This paper proves that hypoelliptic SDEs driven by fractional Brownian motion with H > 1/2 exhibit ergodic behavior similar to classical Brownian motion-driven systems, including uniqueness of stationary solutions.

Contribution

It establishes ergodic properties and the strong Feller property for hypoelliptic SDEs driven by fractional Brownian motion with H > 1/2, extending classical results to this setting.

Findings

Existence of a unique stationary solution for the systems.

Bound on moments of the inverse Malliavin covariance matrix.

Systems satisfy a version of the strong Feller property.

Abstract

We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying H\"ormander's condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that they admit a unique stationary solution that is physical in the sense that it does not "look into the future". The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.