Fractional Brownian motion with deterministic drift: How critical is drift regularity in hitting probabilities

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 non-centered Gaussian process $B^H+f$. It aims to highlight how each component $f$, $E$ and $F$ is involved in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\ne \varnothing \}$. When $F$ is a singleton and $f$ is a general measurable drift, some new estimates are obtained for the last probability by means of suitables Hausdorff measure and capacity of the graph $G{r}_E(f)$. As application we deal with the issue of polarity of points for $(B^H+f){|}_E$ (the restriction of $B^H+f$ to the subset $E\subset (0,\infty )$).