= Newtonian potential of the Brownian heat kernel
{c}
{title2=$\int_0^\infty p_s(x)\,ds=A_d|x|^{2-d}$}
For $d\geq3$, $A_d=\Gamma(d/2-1)/(2\pi^{d/2})$. Substitute $u=|x|^2/(2s)$ to obtain the displayed identity for $x\ne0$. 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.
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