# Particle systems and quasi-stationary distributions

Nothing lasts forever. However many phenomena can be well described by a process which enters a quasi-stationary state before eventually vanishing. On the other hand, using a judicious state representation, many phenomena (e.g. in chemistry, ecology, genetics, population dynamics, telecommunications networks, statistical physics,...) can be appropriately described by Markov processes.

Since the pioneering work of Kolmogorov and Yaglom, a lot of work has been dedicated to understand the quasi-stationary behavior of Markov processes. More precisely, for X a continuous time Markov process on a countable state space, with an absorbing state that we call 0, the conditioned evolution is the distribution of the process conditioned on non-absorption while a quasi-stationary distribution (QSD) is a probability measure that is invariant for the conditioned evolution.

Unlike invariant distributions, QSD are solutions of a non-linear equation and there can be 0, 1 or an inﬁnity of them. Also, they cannot be obtained as Cesàro limits of Markovian dynamics. These facts make the computation of QSDs a nontrivial matter.

We study different particles systems (Branching particles, Branching with selection, Fleming Viot systems) allowing to simulate QSD distributions. We also explain some links with the existence of traveling waves for some specific PDEs.

Joint work with P. Groisman.