Gaussian heat-kernel proof of the harmonic Liouville theorem
= Gaussian heat-kernel proof of the harmonic Liouville theorem
{c}
A bounded entire <harmonic function> satisfies $u(x)=\mathbb E_xu(B_t)$ because $u(B)$ is a bounded <local martingale>. The <Gaussian heat kernel> has a directional derivative with $L^1$ norm $\sqrt{2/\pi}/\sqrt t$. Consequently $|u(x)-u(y)|\leq\|u\|_\infty\sqrt{2/\pi}|x-y|/\sqrt t$; letting $t\to\infty$ proves the <harmonic Liouville theorem> in every dimension.