Maximum of N independent Brownian walkers till the first exit from the half-space

We consider the one-dimensional target search process that involves an immobile target located at the origin and N searchers performing independent Brownian motions starting at the initial positions (x) over right arrow = (x(1), x(2),..., x(N)), all on the positive half-space. The process stops when the target is first found by one of the searchers. We compute the probability distribution of the maximum distance m visited by the searchers till the stopping time and show that it has a power-law tail: P-N(m|(x) over right arrow) similar or equal to B-N(x(1)x(2) ... x(N))/m(N+1) for large m. Thus, all moments of m up to the order (N - 1) are finite, while the higher moments diverge. The prefactor B-N increases with N faster than exponentially. Our solution gives the exit probability of a set of N particles from a box [ 0, L] through the left boundary. Incidentally, it also provides an exact solution of Laplace's equation in an N-dimensional hypercube with some prescribed boundary conditions. The analytical results are in excellent agreement with Monte Carlo simulations.