[zcjaojwe] Point growth differential equation

Generalize the point growth simulation to the continuous case.

Place a seed of arbitrary shape at the center of a circle, and randomly select a start point on the edge of the circle.  The point moves randomly (brownian motion?) until it touches the seed, where it sticks (we'll skip the more complicated tree criterion about the sticking point).  If the point wanders beyond the edge of the initial circle, the simulation restarts from the same initial start point.

What is the probability distribution of all the places the point might stick, that is, the probability over the boundary of the seed?  Then, average the probabilities over all possible initial points on the circle.  Then, take the limit as the initial circle's radius tends to infinity.

I suspect the answer is an application of the Laplacian.  We need to get an answer so that the inside of a "bay" with a narrow inlet is unlikely to be accessed.