Newtonian potential of the Brownian heat kernel (${\int }_0^{\infty }{p}_s(x)ds=A_d|x{|}^{2-d}$)

For $d\ge 3$, $A_d=\Gamma (d/2-1)/(2{\pi }^{d/2})$. Substitute $u=|x{|}^2/(2s)$ to obtain the displayed identity for $x\ne 0$. The semigroup property and Tonelli theorem then give $P_t(|\cdot {|}^{2-d})(x)=A_d^{-1}{\int }_t^{\infty }{p}_s(x)ds$. The coefficient multiplying the time integral is the reciprocal of the coefficient in the full Green integral, a distinction important in Brownian motion hitting asymptotics.