Hitting Probabilities for Hypoelliptic Differential Equations Driven by Fractional Brownian Motion

TL;DR

This paper derives estimates for the likelihood that solutions to hypoelliptic SDEs driven by fractional Brownian motion hit certain sets, extending classical elliptic results to a more general hypoelliptic and fractional context.

Contribution

It establishes joint density smoothness for hypoelliptic SDEs driven by Gaussian rough paths and derives hitting probability estimates in terms of control distance for fractional Brownian motion.

Findings

Established existence and smoothness of joint densities for hypoelliptic SDEs driven by Gaussian rough paths.

Derived local upper bounds for densities in the fractional Brownian motion setting.

Provided two-sided estimates for hitting probabilities using Newtonian-type capacities.

Abstract

The main goal of this article is to derive a two-sided estimate for hitting probabilities of a hypoelliptic stochastic differential equation (SDE) driven by fractional Brownian motion (fBM) with Hurst parameter $H\in(1/4,1)$ in terms of Newtonian-type capacities that are defined with respect to the (sub-Riemannian) control distance associated with the vector fields. As a starting point, we first establish the existence and smoothness of joint densities for the finite-dimensional distributions of the solution in the general context of hypoellitpic SDEs driven by Gaussian rough paths. We then turn to the fBM setting and derive a local upper bound for the joint density in terms of the control distance. As an application of these results, we establish our main estimate on hitting probabilities which generalises a well-known elliptic result of \cite{BNOT} to the hypoelliptic case.