Regularisation by Gaussian rough path lifts of fractional Brownian motions

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 $H\in(1/3,1/2)$.