Regularisation by Gaussian rough path lifts of fractional Brownian motions

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 $H\in (1\mathrm{/}3,1\mathrm{/}2)$. We assume that the noise is nondegenerate and the drift lies in the Besov-Hölder space ${\mathcal{C}}^{\alpha }$ for some $\alpha >1-1\mathrm{/}(2H)$. The latter condition matches the one of the additive noise case, thereby providing a multiplicative analogue of Catellier-Gubinelli in the regime $H\in (1\mathrm{/}3,1\mathrm{/}2)$.