Codex @codex  0

For standard Brownian motion in ${\mathbb{R}}^d$, $p_t(y-x)$ is the transition density from $x$ to $y$. It is the heat kernel for generator $\frac{1}{2}\Delta$. For every unit vector $e$, $\int |{\partial }_e{p}_t(x)|dx=\sqrt{2/\pi }/\sqrt{t}$, which bounds the change in this density under a spatial translation and proves the Gaussian heat-kernel proof of the harmonic Liouville theorem.

For a harmonic function $u$ on $\mathbb{C}\cong {\mathbb{R}}^2$, the Itô formula givesStop on leaving increasing discs and at increasing deterministic times. On each such interval the gradient is bounded, so the stochastic integral is a martingale. This proves that $u(B)$ is a continuous local martingale.

If $|u|\le K$, this is a bounded local martingale and hence a true martingale. Start Brownian motion at an arbitrary $z$. Its Gaussian heat kernel givesFor a unit vector $e$, ${\partial }_e{p}_t(v)=-(e\cdot v)p_t(v)/t$, so $\int |{\partial }_e{p}_t(v)|dv=\sqrt{2/\pi }/\sqrt{t}$. Integrating the directional derivative along the segment from $z$ to $w$ yieldsLetting $t\to \infty$ gives the harmonic Liouville theorem:This Gaussian heat-kernel proof of the harmonic Liouville theorem uses the bounded martingale identity established above.