= Killed geometric Brownian heat kernel
{title2=$p(t,x,y)$}
The <differential operator> $L=\tfrac12x^2\partial_{xx}+\tfrac12x\partial_x-\tfrac12$ has solution operator $P_tg(x)=e^{-t/2}\mathbb E[g(xe^{B_t})]$. For $x\ne0$ its <integral kernel> with respect to <Lebesgue measure> is $p(t,x,y)=e^{-t/2}(|y|\sqrt{2\pi t})^{-1}\exp[-\log^2(|y/x|)/(2t)]$ when $xy>0$, and zero otherwise. For $x=0$ the kernel is the <measure> $e^{-t/2}\delta_0$, not a density. The <Itô formula> applied to $e^{-s/2}u(t-s,X_s)$ proves this <Feynman-Kac formula>.
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