Euclidean heat kernel

For $t>0$ and $x\in\mathbb R^n$ , the Euclidean heat kernel with diffusivity $\nu>0$ is

G_\nu(t,x)=(4\pi\nu t)^{-n/2}\exp\!\left(-\frac{|x|^2}{4\nu t}\right).

It is positive, has integral one in $x$ , and solves the heat equation for positive time.

Normalization and concentration

The Gaussian integral proves the mass normalization, and direct differentiation proves $\partial_tG_\nu=\nu\Delta G_\nu$ . Its scaling is $G_\nu(t,x)=(\nu t)^{-n/2}G_1(1,x/\sqrt{\nu t})$ . For each fixed $\delta>0$ , the integral over $|x|>\delta$ tends to zero as $t\downarrow0$ . The time-zero limit is a point mass in the distributional sense, not an ordinary integrable function.

References

Hunter, The Heat and Schrödinger Equations, §§5.1–5.3.