A first passage time problem for random walk occupancy

Abstract

Two recent studies of diffusion and flow properties of polymers in a melt have suggested the problem of finding the average time form Brownian particles to leave a sphere for the first time, given that exited particles can also reenter the sphere. We prove that the asymptotic density (asm→∞) for the time to first emptiness of the sphere for zero-mean Brownian motion is a delta function, characterized by the exit timea(m/lnm)^2/D,a being a constant andD being the dimension. The presence of a field leaves the delta-function form for the density, but changes the time dependence toa lnm, with only the constanta depending on the dimension. Simulations of the process suggest that the value ofm needed for the validity of the asymptotic result is orders of magnitude greater than 1000.