Properties of the heat kernel

Question

The function $$g_t(x) = \frac{1}{(4\pi t)^{n/2}}\exp\left(-\frac{|x|^2}{4t}\right)~~~\text{for}~~~t>0~~~\text{and}~~~x\in\mathbb R^n$$ denotes the heat kernel. I want to show that $g_{s+t}=g_s*g_t$ for $s,t>0$ where $*$ is the convolution, but I do struggle with what the convolution looks like.

Answer 1

The convolution integral will contain the product of two exponentials, so their arguments are added:

$$\exp(-\frac{|x-y|^2}{4t}-\frac{|y|^2}{4s})$$

Put the two fractions on a common denominator and re-arrange the terms of the numerator so that they form a complete square in $y$ again (plus a constant term).

Answer 2

If you have been told the relationship between convolutions and their Fourier transforms (where convolution corresponds to ordinary point-wise multiplication) then the heat kernel identity is especially easy to deduce.