Meeting time of two independent Brownian motions
= Meeting time of two independent Brownian motions
Two independent one-dimensional Brownian motions started at $-a$ and $a$ meet at the first time a Brownian motion started at $\sqrt2a$ hits zero. The meeting time has density
$$
f_T(u)=\frac{a}{\sqrt{\pi u^3}}e^{-a^2/u},
\qquad u>0.
$$
An orthogonal sum coordinate is independent of this hitting time and makes the meeting position conditionally $N(0,u/2)$.