Phase Transition for Absorbed Brownian Motion with Drift

We study one-dimensional Brownian motion with constant drift toward the origin and initial distribution concentrated in the strictly positive real line. We say that at the Iirst time the process hits the origin, it is absorbed. We study the asymptotic behavior, as t ~ ,:,'., of m,, the conditional distribution at time zero of the process conditioned on survivul tip to time t and on the process having a fixed value at time t. We iliad that there is a phase transition in the decay rate of the initial condition. For fast decay rate (subcritical case) m, is localized, in the critical case m, is located around x/~, and for slow rates (supercritical case) m, is located around t. The critical rate is given by the decay of the mi,fimal qt, asistationary distribution of this process. We also study in each case the asymptotic distribution of the process, scaled by x/7, conditioned as before. We prove that in the subcritical case this distribution is a Brownian excursion. In the critical case it is a Brownian bridge attaining 0 for the first time at time I. with some initial distribution. In the supercritical case, after centering around the expected value--which is of the order of t -we show that this process converges to a Brownian bridge arriving at 0 at time 1 and with a Gaussian initial distribution.