= Half-line drift boundary kernel
{title2=$P_\alpha(x,t)=\frac{x}{2\sqrt\pi t^{3/2}}e^{-(x+\alpha t)^2/(4t)}$}
Convolution of this kernel with <Dirichlet boundary data> supplies the boundary forcing for $u_t=u_{xx}+\alpha u_x$ on $x>0$. It is $-(2\partial_x+\alpha)H(x+\alpha t,t)$. For $\alpha\ge0$, its total time mass is $e^{-\alpha x}$, tending to one as $x\downarrow0$, and its mass away from zero time tends to zero. Thus the boundary value is recovered through an <approximate identity>, not by pointwise substitution $x=0$ in the kernel.
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