Analysis of a Brownian particle moving in a time‐dependent drift field

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)≥0). We assume that Y(T₀)=X₀≥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)∈(X,X+ dX)|Y(T₀)=X₀≥0], in terms of Airy functions. We then obtain detailed asymptotic results, that apply for X₀ and/or T₀→∞, and various ranges of the space–time (X,T) plane. We interpret our results in terms of semi‐classical mechanics.