Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 4 / c / Solution

Apply the orthogonal transformationThe processes $D$ and $C$ are independent one-dimensional Brownian motions, with $D_0=\sqrt{2}a$ and $C_0=0$. The meeting time is the first time $D$ hits zero, which is almost surely finite by one-dimensional Brownian recurrence.

The Brownian reflection principle gives the first-passage density from $x>0$ to zero asSubstituting $x=\sqrt{2}a$ gives the meeting time of two independent Brownian motions density