Where does a random process hit a fractal barrier?

TL;DR

This paper investigates the hitting times of a Brownian motion with respect to a fractal barrier, revealing a singular set of times where the process almost surely hits the barrier, and explores related hitting measure problems.

Contribution

It introduces the concept of a singular time set for Brownian paths hitting fractal barriers and addresses new problems in hitting measure theory for random processes.

Findings

Existence of a singular time set for Brownian hitting times

Almost sure hitting of the barrier at times in the singular set

Addresses new problems in hitting measure for stochastic processes

Abstract

Given a Brownian path $\beta(t)$ on $\mathbb{R}$, starting at $1$, a.s. there is a singular time set $T_{\beta}$, such that the first hitting time of $\beta$ by an independent Brownian motion, starting at $0$, is in $T_{\beta}$ with probability one. A couple of problems regarding hitting measure for random processes are presented.