Where does a random process hit a fractal barrier?

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

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