Precise Local Estimates for Differential Equations driven by Fractional Brownian Motion: Hypoelliptic Case

TL;DR

This paper develops precise local estimates for solutions of hypoelliptic stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/4, using rough path theory.

Contribution

It provides sharp local control distance and density estimates for hypoelliptic equations driven by fractional Brownian motion, advancing understanding of their regularity properties.

Findings

Established sharp local control distance estimates.

Derived lower bounds on the solution density.

Utilized rough path techniques for analysis.

Abstract

This article is concerned with stochastic differential equations driven by a $d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$, 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.