Consider the problem of n predators X_1,...,X_n chasing a single prey X_0, all independent standard Brownian motions on the real line. If the prey starts to the right of the predators, X_k(0) < X_0(0) for all k=1,...,n, then the first capture time is
T_n = inf{ t > 0 : X_0(t) = X_k(t) for some k }. Equivalently, this is the first exit time for a Brownian motion that starts at an interior point of the corresponding cone in R^(n+1). Bramson and Griffeath (1991) showed that the expected capture time
E(T_n) = ? for n = 1, 2, 3 and, based on simulation, conjectured that E(T_n) < ? for n > 4. This conjecture was proved by W. V. Li and Q. M. Shao (2001) for n > 4 using a result of de Blassie (1987), that the finiteness of expectation is equivalent to a specific lower bound of the first Dirichlet eigenvalue of the domain which is the intersection of cone with the unit n-dimensional sphere at the origin.
I will discuss my joint work with J. Ratzkin, in which we prove the conjecture for n = 4 by establishing the eigenvalue estimate.