Where does a random process hit a fractal barrier?

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.