Hitting probabilities for fractional Brownian motion with deterministic drift

Abstract

Let $B^H$ be a $d$-dimensional fractional Brownian motion with Hurst index $H\in (0,1)$, $f:[0,1]⟶{\mathbb{R}}^d$ a Borel function, and $E\subset [0,1]$, $F\subset {\mathbb{R}}^d$ are given Borel sets. The focus of this paper is on hitting probabilities of the fractional Brownian motion $B^H$ with the deterministic drift $f$. It aims to highlight the role of the regularity properties of the drift $f$ as well as that of the dimension of $E$ in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\ne \varnothing \}$ for $F$ a subset of ${\mathbb{R}}^d$ and also for $F$ a singleton.