Hitting probabilities for fractional Brownian motion with deterministic drift

TL;DR

This paper investigates the probability that a fractional Brownian motion with a deterministic drift hits a set, emphasizing how the drift's regularity and the domain's dimension influence these probabilities.

Contribution

It provides new bounds on hitting probabilities for fractional Brownian motion with deterministic drift, highlighting the impact of drift regularity and domain dimension.

Findings

Derived upper and lower bounds for hitting probabilities.

Showed the influence of drift regularity on hitting behavior.

Analyzed the role of domain dimension in hitting probabilities.

Abstract

Let $B^{H}$ be a $d$-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f:[0,1]\longrightarrow\mathbb{R}^{d}$ 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 $\mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}$ for $F$ a subset of $\mathbb{R}^{d}$ and also for $F$ a singleton.