Consider a Brownian particle in the plane with a circular trap at the origin. If we give the particle enough time it falls into the trap (since Brownian motion is space filling in 2D). However, suppose we only let the particle evolve for a finite interval of time. What is the probability it is trapped?
Scaling time and space appropriately we may assume that the trap is the unit disk. Formulated precisely the problem is. Start a standard Brownian motion in the plane at some distance $r>1$ from the origin. Let it run up to time $T$. What is the probability $p(r,T)$ that it hits the unit disk at some time $t\in [0,T]$?
This can, of course, be rephrased entirely in terms of a boundary value problem:
$$\frac{\partial p}{\partial T}= \frac{1}{2r} \frac{\partial }{\partial r} r \frac{\partial p}{\partial r},$$
$$p(1,T)=1, \quad p(r,0)=0, \ 1 \lt r \lt \infty, \quad \ \lim_{r \rightarrow \infty} p(r,T)=0.$$
If we take $T\rightarrow \infty$ then $p(r,T)\rightarrow 1$ for every $r$, but for any fixed $T<\infty$, $p(r,T)<1$ and decays as $r\rightarrow \infty$ at least like a Gaussian.
Surely this has been studied somewhere in the literature. My question is, "Where?" Maybe even some sort of "exact" formula exists for $p(r,T)$ involving special functions — probably Bessel functions. I am particularly interested in understanding the "effective area of the trap", i.e., the area of the disk over which $p(r,T)$ is "almost 1", say $>1-\epsilon$ for some fixed $\epsilon$, but any reference to a study of this type of finite time hitting problem would be appreciated.
Best Answer
Instead of a concrete answer, I will give what appears to be the most useful reference. I quote the first paragraph of Wendel, J. G. "Hitting spheres with Brownian motion". Ann. Prob. 8, 164 (1980).
See also Betz, C. and Gzyl, H. "Hitting spheres from the exterior". Ann. Prob. 22, 177 (1994) and various cites of these papers.