Konstantinos Dareiotis (University of Leeds UK)

Regularisation of differential equations by multiplicative fractional noises

Abstract In this talk, we consider differential equations perturbed by multiplicative fractional Brownian noise. Depending on the value of the Hurst parameter H, the resulting equation is pathwise viewed as an ordinary (H>1), Young (H in (1/2, 1)) or rough (H in (1/3, 1/2)) differential equation. In all three regimes we show regularisation by noise phenomena by proving the strongest kind of well-posedness for equations with irregular drifts: 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 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. This is a joint work with Máté Gerencsér.