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.


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).

Let $X_t$ be a standard $d$-dimensional Brownian motion with nonrandom starting point $X_0$. When $d \ge 2$ we seek explicit formulas which will determine the joint distributions of the first time $T \le \infty$ and place $X_T$ (which is only defined when $T$ is finite) where $X_T$ hits a sphere centered at the origin, either from the inside or from the outside, or exits from the region bounded by concentric spheres.

See also Betz, C. and Gzyl, H. "Hitting spheres from the exterior". Ann. Prob. 22, 177 (1994) and various cites of these papers.

  • $\begingroup$ Thanks! That is pretty much exactly what I was looking for, down to the exact formulas involving Bessel functions! $\endgroup$ – Jeff Schenker May 12 '12 at 5:28

See here. Also, for a much more general framework than that of heat/diffusion, see "Stopping time" at the wikipedia.

  • 2
    $\begingroup$ See what there? Can you give more detail? How does your link answer the question? The wikipedia link contains the definition of stopping times, but I don't think it tells you the distribution of the stopping time asked for here. $\endgroup$ – George Lowther May 11 '12 at 21:54

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