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

2021/12/03 by Hakiki, Youssef, Erraoui, Mohamed
#FOS: Mathematics #Primary 60J65 #Probability (math.PR)

paper · doi:10.48550/arxiv.2112.02055

Abstract

Let B^H be a fractional Brownian motion in ℝ^d of Hurst index H∈(0,1), f:[0,1]\longrightarrowℝ^d a Borel function and A⊂[0,1] 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.

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