Probability of Brownian particle absorption

[alex_rodin]

Homework Statement

There is a brownian particle in 3D space and absorbing sphere with radius a. At moment t = 0 the particle was situated at distance l from the sphere. Caluclate the probability of absorbing the particle by the sphere.

Homework Equations

n is probability densithy for the particle,
$$\partial _t n = D \Delta n$$

The Attempt at a Solution

The equation for probability density for the particle in spherical coordinates is
$$\partial _t n = D \frac 1 {r^2} \partial _r \left(r^2 \partial _r n\right)$$
with initial conditions
$$n(r,0) = \frac {\delta \left(r - r_0\right)} {4 \pi r_0^2}, \ r_0 = a + l$$

But what is the right boundary conditions for the equation?

Also, what is the right way to calculate the probability of absorbtion?

 

[haruspex]

Can you obtain a generic solution of the PDE using e.g. separation of variables?