Gaussian heat kernel
= Gaussian heat kernel
{c}
{title2=$p_t(x)=(2\pi t)^{-d/2}e^{-|x|^2/(2t)}$}
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 $\tfrac12\Delta$. 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>.