Coordinatewise coalescing coupling of Brownian motions
= Coordinatewise coalescing coupling of Brownian motions
Two Brownian motions in $\mathbb R^d$ can be coupled by letting corresponding coordinates evolve independently until they meet and then using the same increments in that coordinate. One-dimensional recurrence makes every coordinate meeting time finite almost surely, so the two processes eventually coalesce while each marginal remains Brownian.