The heat kernel on ${\mathbb{R}}^n$ and its causal extension

Let $n\ge 1$ and $t>0$. The heat kernel on ${\mathbb{R}}^n$ is the function

$\mathrm{\Gamma }(x,t):=(4\pi t{)}^{-n\mathrm{/}2}\exp \text{\hspace{0.17em}⁣}(-\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{4t}),x\in {\mathbb{R}}^n,$

where $\mathrm{\mid }x{\mathrm{\mid }}^2=⟨x,x⟩$ is the Euclidean square of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and $\exp$ is the real exponential function of The real exponential function and the number $e$ by a power series. For fixed $t>0$ the map $x\mapsto \mathrm{\Gamma }(x,t)$ is a composite of the quadratic form with the exponential and a scalar multiple, so it is $C^{\mathrm{\infty }}$ in the sense of $C^k$ maps and multi-index derivative notation in Euclidean space by The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$ and $C^k$ Euclidean maps are closed under componentwise algebra and composition, and it is strictly positive by The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$. The spatial Laplacian entering the heat equation is that of The Laplacian of a $C^2$ function and of a $C^2$ vector field.

The causal extension of the heat kernel is $\mathrm{\Gamma }(x,t)$ for $t>0$ and $\mathrm{\Gamma }(x,t):=0$ for $t\le 0$; the displayed formula is not evaluated at $t=0$ as a function value. The normalisation is fixed by the unit-mass identity proved for the kernel on this page, and the causal extension is used only as a locally integrable function or, after embedding, as a distribution.