On probability laws of solutions to differential systems driven by a fractional Brownian motion

Abstract

This article investigates several properties related to densities of solutions $(X_{t})_{t\in[0,1]}$ to differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/4$. We first determine conditions for strict positivity of the density of $X_{t}$. Then we obtain some exponential bounds for this density when the diffusion coefficient satisfies an elliptic type condition. Finally, still in the elliptic case, we derive some bounds on the hitting probabilities of sets by fractional differential systems in terms of Newtonian capacities.