Capture time of Brownian pursuits

τn = inf{t > 0 : Bi(t)− bi = B0(t) for some 1 ≤ i ≤ n}, where 0 < bi ≤ 1, 1 ≤ i ≤ n. The τn can be viewed as a capture time in a random pursuit setting. Assume that a Brownian prisoner escapes, running along the path of B0. In his pursuit, there are n independent Brownian policemen. These policemen run along the paths of B1, · · · , Bn, respectively. At the outset, the prisoner is ahead of the policemen by some fixed distances bi, 1 ≤ i ≤ n. Then, τn represents the capture time when the fastest of the policemen catches the prisoner. In an elegant paper on coupling various stochastic processes, Bramson and Griffeath (1991) considered the analogous stopping time τ̃n for continuous time random walks. It is very likely that the kind of tail estimates which we derive here for τn are the same for τ̃n. However, for our purposes Brownian motions are easier to work with, so that we will stick with the setup described above. Bramson and Griffeath raised the question: For which n is τn < ∞. A more animated interpretation is “How many Brownian policemen does it take to arrest a Brownian prisoner?” They showed for continuous time random walks that τ̃n = ∞ for n = 2 or 3, and their computer simulations indicated that τ̃n < ∞ for n ≥ 4.