# Phase Transition for Absorbed Brownian Motion with Drift

## P. A. Ferrari, H. Kesten and S. Mart\'{\i}nez

We prove that certain (discrete time) probabilistic automata which can be absorbed in a null state'' have a normalized quasi-stationary distribution (when restricted to the states other than the null state). We also show that the conditional distribution of these systems, given that they are not absorbed before time $n$, converges to an honest probability distribution; this limit distribution is concentrated on the configurations with only finitely many active or occupied'' sites.

A simple example to which our results apply is the discrete time version of the subcritical contact process on $\Z^d$ or oriented percolation on $\Z^d$ (for any $d\ge 1$) as seen from the leftmost particle''. For this and some related models we prove in addition a central limit theorem for $n^{-{1\over 2}}$ times the position of the leftmost particle (conditioned on survival till time $n$).

The basic tool is to prove that our systems are $R$-positive-recurrent. \endabstract

\keywords Absorbing Markov chain. Quasi-stationary distribution. Ratio limit theorem. Yaglom limit. $R$-positivity. Central limit theorem. \endkeywords

\subjclass Primary 60J10, 60F05 Secondary 60K35