Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 34 / 4 / b

To keep the normalization visible, denote the integral defining the printed constant by $A_d$. Substituting $r=|y{|}^2/(2s)$ gives, for $y\ne 0$,This is the Newtonian potential of the Brownian heat kernel. The singularity at zero is locally integrable in dimension $d$, since its radial integral behaves like ${\int }_0^1{r}^{2-d}{r}^{d-1}dr=1/2$. All integrands are nonnegative, so Tonelli's theorem permits both orders of integration.

Integrating in $s$ first givesIntegrating in $y$ first uses the Gaussian heat kernel convolution. Completing the square in $|y-x{|}^2/t+|y{|}^2/s$ givesConsequently $I={\int }_t^{\infty }p(r,0,x)dr$, and the actual identity isThe original PDF uses $A_d$ rather than $A_d^{-1}$ as the multiplier while defining its constant to equal $A_d$. That formula is false. For example $A_3=1/(2\pi )$, so the required multiplier in dimension three is $2\pi$. Equivalently, as $t\downarrow 0$ at $x\ne 0$, the left side tends to $|x{|}^{2-d}$, whereas the printed right side tends to $A_d^2|x{|}^{2-d}$. One can retain the printed form only by defining its multiplier as $c_d=A_d^{-1}$.