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

TL;DR

This paper investigates conditions under which the image of a fractional Brownian motion with deterministic drift has positive Lebesgue measure or a non-empty interior, based on properties of the drift function and the set involved.

Contribution

It provides new sufficient conditions linking the Hausdorff dimension of the drift's graph and the set to the measure and interior properties of the image.

Findings

If the parabolic Hausdorff dimension of the graph of f exceeds Hd, the density of the occupation measure is square integrable.

If the Hausdorff dimension of A exceeds Hd, the density admits a continuous version.

These results establish criteria for the image of fractional Brownian motion with drift to have positive measure or interior.

Abstract

Let $B^{H}$ be a fractional Brownian motion in $\mathbb{R}^{d}$ of Hurst index $H\in\left(0,1\right)$, $f:\left[0,1\right]\longrightarrow\mathbb{R}^{d}$ 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.