Path-by-path regularisation through multiplicative noise in rough, Young, and ordinary differential equations

Differential equations perturbed by multiplicative fractional Brownian motions are considered. Depending on the value of the Hurst parameter $H$, the resulting equation is pathwise viewed as an ODE, YDE, or RDE. In all three regimes we show regularisation by noise phenomena by proving the strongest kind of well-posedness with irregular drift: strong existence and path-by-path uniqueness. In the Young and smooth regime $H>1/2$ the condition on the drift coefficient is optimal in the sense that it agrees with the one known for the additive case [CG16, Ger22]. In the rough regime $H\in(1/3,1/2)$ we assume positive but arbitrarily small drift regularity for strong well-posedness, while for distributional drift we obtain weak existence.