 Brownian coupling proof of the harmonic Liouville theorem

For a bounded harmonic function $f$ on ${\mathbb{R}}^d$, the processes $f(B_t^x)$ and $f(B_t^y)$ are bounded martingales. Couple the Brownian motions from $x$ and $y$ so that they coalesce in finite time almost surely. Thenwhich proves that $f$ is constant in every dimension.