Brownian coupling proof of the harmonic Liouville theorem
= Brownian coupling proof of the harmonic Liouville theorem
{c}
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. Then
$$
|f(x)-f(y)|
\leq2\lVert f\rVert_\infty\mathbb P(T>t)
\longrightarrow0,
$$
which proves that $f$ is constant in every dimension.