Smooth density for some nilpotent rough differential equations

TL;DR

This paper presents an example of a rough differential equation driven by fractional Brownian motion with Hurst parameter between 1/3 and 1/2, showing the solution has a smooth density using explicit solution representation and a Norris type lemma.

Contribution

It provides a novel example of a rough differential equation with a smooth density, leveraging nilpotent vector fields and rough paths techniques.

Findings

Solution admits a smooth density with respect to Lebesgue measure.

Explicit representation of the solution is constructed for nilpotent vector fields.

A Norris type lemma in the rough paths context is established.

Abstract

In this note, we provide a non trivial example of differential equation driven by a fractional Brownian motion with Hurst parameter 1/3 < H < 1/2, whose solution admits a smooth density with respect to Lebesgue's measure. The result is obtained through the use of an explicit representation of the solution when the vector fields of the equation are nilpotent, plus a Norris type lemma in the rough paths context.