Phase Transition for Absorbed Brownian Motion with Drift.

We study one dimensional Brownian motion with constant drift towards the origin and initial distribution concentrated in the strictly positive real line. We say that at the rst time the process hits the origin, it is absorbed. We study the asymptotic behavior, as t!1, of m t, the conditional distribution at time zero of the process conditioned on survival up to time t and on the process having a xed value at time t. We nd that there is a phase transition in the decay rate of the initial condition. For fast decay rate (subcritical case) m t is localized, in the critical case m t is located around