Heat Poisson kernel
= Heat Poisson kernel
{c}
{title2=$P(x,t)$}
For $x>0$ and the <heat equation> on a half-line, the zero-initial-data solution with prescribed <Dirichlet boundary condition> $g$ is $\int_0^tP(x,t-s)g(s)ds$, where
$$
P(x,t)=\frac{x}{2\sqrt\pi\,t^{3/2}}e^{-x^2/(4t)},\qquad\mathcal L_tP=e^{-x\sqrt p}.
$$
Its mass concentrates at $t=0$ as $x\downarrow0$, recovering the boundary trace.