Images of Fractional Brownian motion with deterministic drift: Positive Lebesgue measure and non-empty interior

Abstract

Let $B^H$ be a fractional Brownian motion in ${\mathbb{R}}^d$ of Hurst index $H\in \left(0,1\right)$, $f:\left[0,1\right]⟶{\mathbb{R}}^d$ a Borel function and $A\subset \left[0,1\right]$ a Borel set. We provide sufficient conditions for the image $(B^H+f)(A)$ to have a positive Lebesgue measure or to have a non-empty interior. This is done through the study of the properties of the density of the occupation measure of $(B^H+f)$. Precisely, we prove that if the parabolic Hausdorff dimension of the graph of $f$ is greater than $Hd$, then the density is a square integrable function. If, on the other hand, the Hausdorff dimension of $A$ is greater than $Hd$, then it even admits a continuous version. This allows us to establish the result already cited.