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 , 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 > 1/2 the condition on the drift coefficient is optimal in the sense that it agrees with the one known for the additive case CG , Ger .In the rough regime ∈ (1/3, 1/2) we assume positive but arbitrarily small drift regularity for strong well-posedness, while for distributional drift we obtain weak existence.This is the point where the shifted variant of the stochastic sewing lemma is used, otherwise the power ( -2) would need to be integrated, which would result in the "wrong" condition > 2 -1/ .