What is the kernel for reflected Brownian motion at some lower barrier $p_b$?
The best I could come up with is:
$(e^{-((x-u)/a)^2/2}+e^{-((x+u-2b)/a)^2/2)})/(2\pi)^{1/2}$
Which is equal to zero when the derivative is taken and $x=p_b$
But this differs from what Wikipedia's entry for a barrier at b=0
http://en.wikipedia.org/wiki/Reflected_Brownian_motion
$\Phi \left(\frac{z-\mu t}{\sigma t^{1/2}} \right) - e^{2 \mu z /\sigma^2} \Phi \left( \frac{-z-\mu t}{\sigma t^{1/2}} \right)$
I have no idea where $ e^{2 \mu z /\sigma^2}$ arrises> I have tried all combination of integrals and can't get that exponent in front.
Wipipedia has no link to how they derived it, and I don;t know if it would work for an arbitrary barrier. Also Do't know what the original distribution is since erf is an integral, so it had to come from some combination of exponential functions