Precise local estimates for differential equations driven by fractional Brownian motion: Hypoelliptic case

Abstract

This article is concerned with stochastic differential equations driven by a d dimensional fractional Brownian motion with Hurst parameter $\mathit{H}>1/4$ and understood in the rough paths sense. Whenever the coefficients of the equation satisfy a uniform hypoellipticity condition, we establish a sharp local estimate on the associated control distance function and a sharp local lower estimate on the density of the solution. Our methodology relies heavily on the rough paths structure of the equation.