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. In the rough regime H ∈ (1/3, 1/2), we assume positive but arbitrarily small drift regularity for strong well-posedness, while for distributional drift we obtain weak existence.