  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.
