PHASE-TRANSITION FOR ABSORBED BROWNIAN-MOTION WITH DRIFT

We study one-dimensional Brownian motion with constant drift toward th e origin and initial distribution concentrated in the strictly positiv e real line. We say that at the first time the process hits the origin , it is absorbed We study the asymptotic behavior, as t --> infinity, of m(t), the conditional distribution at time zero of the process cond itioned on survival up to time t and on the process having a fixed val ue at time t. We find that there is a phase transition in the decay ra te of the initial condition. For fast decay rate (subcritical case) m( t) is localized, in the critical case m, is located around root t, and for slow rates (supercritical case) m(t) is located around t. The cri tical rate is given by the decay of the minimal quasistationary distri bution of this process. We also study in each case the asymptotic dist ribution of the process, scaled by root t, conditioned as before. We p rove that in the subcritical case this distribution is a Brownian excu rsion. In the critical case it is a Brownian bridge attaining 0 for th e first time at time 1, with some initial distribution. In the supercr itical case, after centering around the expected value-which is of the order of t-we show that this process converges to a Brownian bridge a rriving at 0 at time 1 and with a Gaussian initial distribution.