I place two spherical particles, $P\_1$ and $P\_2$ (with radii $r\_1$ & $r\_2$), into a cylindrical container of radius $r\_c$ ($r\_1$ & $r\_2$ $\leq \frac{1}{2}r\_c$) and height $h$. While $P\_1$ is immobilized at the centerpoint of the cylindrical container, $P\_2$ has a coefficient of diffusion $D$, and can freely diffuse throughout the container and across its walls (i.e. the boundaries of the cylindrical container are non-reflecting). 


We randomly position $P\_1$ and $P\_2$ somewhere inside the cylindrical container. Can we derive an expression for the mean residency time of $P\_2$ as a function of the relative sizes of the particles and the cylindrical container? Is it a fair approximation to simply estimate the residency time of $P\_2$ in a cylindrical container resized to subtract the volume of $P\_1$?


Update - $P\_1$ is now fixed at the centerpoint of the cylindrical container, and $r\_1$ & $r\_2$ are defined to be $\leq \frac{1}{2}r\_c$, s.t. $P\_1$ cannot block access, say, to half the cylindrical container. 


Update 2 - In practice, $r\_1$, $r\_2$, and $r\_c$ will be within one or two orders of magnitude of one another, i.e. it is not the case that $r\_1$ & $r\_2$ $<< \frac{1}{2}r\_c$). Also, all collisions between particles, like the walls of the container, are fully reflecting.


Update 3 - I'd be perfectly happy calling the cylindrical container a "spherical container" with the same radius, $r\_c$. Similarly, I'd be happy to not pin $P\_1$ at any particular point, and instead to simply make the walls of the sphere reflecting specifically for this particle. 



Residency time of a spherical Brownian particle in a cylindrical container with another spherical particle at a fixed position
You can use the [Feynman-Kac](http://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula) formula to get the [Moment Generating Function](http://en.wikipedia.org/wiki/Moment-generating_function) of the time it takes the particle to leave.


I will consider the case where you fix $P\_1$ and let $P\_2$ move. Whatever the geometry of your problem, you can get an equivalent problem with a point particle diffusing in some region in space, where there is one wall that it reflects off (let's call it $\Gamma\_0$) and another where it is absorbed (let's call it $\Gamma\_1$.) Let $X(t)$ be the position of the particle at time $t$.
Let $T$ be the time at which the process first hits $\Gamma\_1$. Let $f(x,\lambda) =E [ e^{\lambda T} | X(0)=x]$. Then Feynman-Kac gives you that $f$ satisfies
$\nabla^2 f/2 + \lambda f =0$
with $\frac{\partial f}{\partial n} = 0$ on $\Gamma\_0$ and $f=1$ on $\Gamma\_1$.
This is the Helmholtz equation on your domain with mixed boundary conditions.
$f(x,\lambda)$ is the moment generating function of $T$ with $X(t)=x$, evaluated at $\lambda$. So now you can compute the mean of $T$, etc.


One geometry for your problem for which you can get an exact solution for each $\lambda$ is if you have one sphere fixed at the middle of the big sphere and you are tracking a point particle that diffuses, bounces off the small sphere and is absorbed by the big sphere. In that case the solutions to the PDE are [spherical Bessel functions](http://en.wikipedia.org/wiki/Bessel_function#Spherical_Bessel_functions%3a_jn.2C_yn).


