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

2023/06/19 by Erraoui, Mohamed, Hakiki, Youssef
#28A78 #60G17 #60J45 #62134 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2306.10922

Abstract

Let B^H be a d-dimensional fractional Brownian motion with Hurst index H∈(0,1), f:[0,1]\longrightarrowℝ^d a Borel function, and E⊂[0,1], F⊂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 ℙ\(B^H+f)(E)∩ F≠ ∅ \. 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 Gr_E(f). As application we deal with the issue of polarity of points for (B^H+f)\vert_E (the restriction of B^H+f to the subset E⊂ (0,∞)).

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