Abstract
We consider a Brownian particle that is subject to (1) a time-dependent convection (or drift) field and (2) a reflecting barrier. We let Y(T) be the particle's position at time T. There is a standard reflecting barrier that constrains the particle to the non-negative real axis (i.e., Y(T) greater than or equal to 0). We assume that Y(T-0) = X-0 greater than or equal to 0 with probability one, and that the drift field is linearly dependent upon time. Specifically, we assume that the drift changes sign at T = 0 and becomes positive for T > 0. Such models arise naturally in several areas, including convection-diffusion problems in mathematical physics and the study of time dependent queues. We obtain an exact expression for the probability density Q, with Q(X, T) dX = Prob[Y(T) is an element of (X, X + dX) / Y(T-0) = X-0 greater than or equal to 0], in terms of Airy functions. We then obtain detailed asymptotic results, that apply for X-0 and/or T-0 --> infinity, and various ranges of the space-time (X, T) plane. We interpret our results in terms of semi-classical mechanics.