# Birth and death processes on certain random trees: Classification and stationary laws

1 PREVAL - Probability, modelling and evaluation of information processing systems Inria Paris-Rocquencourt 2 LLRS - Laboratory of Large Random Systems

Abstract : The main substance of the paper concerns the growth rate and the classification ergodicity, transience of a family of random trees. In the basic model, new edges appear according to a Poisson process of parameter $\lambda$ and leaves can be deleted at a rate $\mu$. The main results lay the stress on the famous number $e$. A complete classification of the process is given in terms of the intensity factor $ho=\lambda-\mu\,$: it is ergodic if $ho\leq e^{-1}$, and transient if $ho>e^{-1}$. There is a phase transition phenomenon: the usual region of null recurrence in the parameter space here does not exist. This fact is rare for countable Markov chains with exponentially distributed jumps. Some basic stationary laws are computed, e.g.~the number of vertices and the height. Various bounds, limit laws and ergodic-like theorems are obtained, both for the transient and ergodic regimes. In particular, when the system is transient, the height of the tree grows linearly as the time $t\to\infty$, at a rate which is explicitly computed. Some of the results are extended to the so-called multiclass model.

Keywords : Random trees ergodicity transience nonlinear differential equations phase transition

Author: Guy Fayolle - Maxim Krikun - Jean-Marc Lasgouttes -

Source: https://hal.archives-ouvertes.fr/