Regularisation by Gaussian rough path lifts of fractional Brownian motions

TL;DR

This paper proves strong well-posedness for rough differential equations driven by Gaussian rough paths of fractional Brownian motion with Hurst parameter between 1/3 and 1/2, even with distributional drifts.

Contribution

It extends the theory of rough differential equations to include distributional drifts driven by fractional Brownian motion with Hurst parameter in (1/3, 1/2), providing a multiplicative analogue of previous additive noise results.

Findings

Established strong well-posedness under specified conditions.

Extended rough path theory to fractional Brownian motion with Hurst H in (1/3, 1/2).

Demonstrated robustness of solutions with distributional drifts.

Abstract

The aim of the paper is to show the probabilistically strong well-posedness of rough differential equations with distributional drifts driven by the Gaussian rough path lift of fractional Brownian motion with Hurst parameter $H\in(1/3,1/2)$. We assume that the noise is nondegenerate and the drift lies in the Besov-H\"older space $\mathcal{C}^\alpha$ for some $\alpha>1-1/(2H)$. The latter condition matches the one of the additive noise case, thereby providing a multiplicative analogue of Catellier-Gubinelli in the regime $H\in(1/3,1/2)$.