Image Systems for Brownian Motion with a Reflecting Boundary

# Question about Reflecting Barriers


# Question about Reflecting Barriers












I am reading "Reflecting Barriers" related chapters in Cox and Miller's book "The theory of stochastic processes". On page 224 it says that:

$"$ Consider a Wiener process $X(t)$ with drift $\mu$ and variance parameter $\sigma^2$ taking place on the positive half-line where the origin is a reflecting barrier. We suppose that $X(0)=x_0>0$. the p.d.f. of $X(t)$ satisfies the forward equation \begin{equation} \frac{1}{2}\sigma^2\frac{\partial^2 p}{\partial x^2} - \mu \frac{\partial p}{\partial x} = \frac{\partial p}{\partial t} \end{equation} For convenience we again write $p(x_0,x;t) = p(x,t)$. We then have the initial condition \begin{equation} p(x;t) = \delta(x-x_0) \end{equation} and the boundary condition \begin{equation} \left[\frac{1}{2}\sigma^2\frac{\partial p}{\partial x} - \mu p \right]_{x=0} = 0 \end{equation} We may again use the method of images. It is known from the theory of partial differential equations that the appropriate image system for this problem consists of a point image at $x=-x_0$ and a continuous system of images in the range $x\leq -x_0$. Accordingly we take \begin{equation} p(x;t) = \frac{1}{\sigma\sqrt{2\pi t}}\left[e^{-\frac{(x-x_0-\mu t)^2}{2\sigma^2 t}} + A e^{-\frac{(x+x_0-\mu t)^2}{2\sigma^2 t}} + \int^{-x_0}_{-\infty}e^{-\frac{(x+\xi-\mu t)^2}{2\sigma^2 t}}k(\xi) d\xi \right]" \end{equation} Can anyone explain to me, why the structure of $p(x;t)$ should look like that? Can I derive it from the "reflection equality"? Why only care about $x\leq -x_0$, not $-x_0\leq x \leq 0?$ I checked related chapter in its reference book (Sommerfeld 1949)

But I still don't have a clue.