Minimum Mean Hitting Times of Brownian Motion with Constrained Drift

Brownian motion with state-dependent drift in a disk is considered, with reection at the boundary and initial state uniformly distributed over the disk. The mean time to hit the center of the disk depends on the drift eld. The problem considered is to minimize the mean time to hit the center, or a small disk around it, subject to a bound on the integral of the magnitude of the drift vector eld. The problem is inspired by the problem of placing a limited amount of routing information in an enormous packet communication network, in order to minimize the mean time to deliver a packet. A related problem is to minimize the mean time to exit a square. It is demonstrated that angularly symmetric drift elds, with drift pointing directly towards the origin, are not always optimal. If the radius of the disk and the constraint on the total integral of drift are sucien tly large, then a strictly smaller mean hitting time is given by a drift eld with a bicycle wheel structure, with strong drift concentrated on the spokes of the wheel. Paper to be presented at the 27th Conference on Stochastic Processes and their Applications (SPA’27), Cam