Amine Asselah (Créteil), Nathanaël Berestycki (Cambridge), Alexandre Gaudillière (Marseille), Ilie Grigorescu (Miami), Pablo Groisman (Buenos Aires), Claude Le Bris (Marne la Vallée), Jorge Littin (Santiago), Denis Villemonais (Palaiseau) .

We consider a system of asymmetric independent random walks. We fix a pattern A, and denote by T the hitting time of A. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of the dynamics at time t conditioned on {T>t}, at all times. This provides bounds on the rate of convergence of the law conditioned on {T>t} towards its limiting probability measure as time tends to infinity. This is one instance, where the generator is not compact, but where we obtain a bound on the second Dirichlet eigenvalue. This result was obtained in a 2006 paper in AOP, jointly with Pablo Ferrari.

Nathanaël Berestycki (Cambridge): Branching Brownian motion with absorption.

I will discuss asymptotic behaviour of one-dimensional branching Brownian motion with negative drift and absorption at zero, in the near-critical regime where the number of particles stays roughly constant of order N (tending to infinity) for long periods of time. We show that the genealogical structure of the system is governed in the limit by a universal object called the Bolthausen-Sznitman coalescent, on time scales of order (log N)^3. This validates nonrigorous predictions by Brunet, Derrida, Muller and Munier for a closely related fixed-size population model. A key argument in our proof is a quasistationary equilibrium for the density of particles, which seems to be closely related to predicted behaviour for Fleming-Viot processes. Joint work with J. Berestycki and J. Schweinsberg.

Alexandre Gaudillière (Marseille): Quasi-stationary measures and metastability.

We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypothesis for (families of) Markov chains on finite configuration space in some asymptotic regime, including the case of configuration space size going to infinity. By comparing restricted ensemble and quasi-stationary measure, we study point-wise convergence velocity of Yaglom limits and prove asymptotic exponential exit law. We introduce soft measures as interpolation between restricted ensemble and quasi-stationary measure to prove an asymptotic exponential transition law on a generally different time scale. By using potential theoretic tools we prove a new general Poincaré inequality and give sharp estimates via two-sided variational principles on relaxation time as well as mean exit time and transition time. We also establish local thermalization on a shorter time scale and give mixing time asymptotics up to a constant factor through a two-sided variational principal. All our asymptotics are given with explicit quantitative bounds on the corrective terms. This is a joint work with Alessandra Bianchi.

Ilie Grigorescu (Miami): Ergodic properties of some catalytic particle systems.

We discuss a general class of stochastic processes obtained from a given Markov process whose behavior is modified upon contact with a catalyst. An extreme example is when the process is killed, but we are interested in the case when it is updated by redistribution to a new configuration and restarts afresh its underlying evolution.
Examples include particle systems such as the Fleming-Viot process, a Bak-Sneppen type fitness model and processes with rebirth. The scaling limit of the empirical measure for the F-V process is related to the existence of the quasi stationary distribution of the driving process.
In the F-V case for diffusions with hard catalyst (equal to the boundary of an open set) one of the most difficult questions is whether the system is properly defined (i.e. non-explosive), which is proven for non smooth domains, including Lipschitz.
It is also shown that, with probability one, exactly one ancestry line survives for all times.
Joint work with Min Kang from North Carolina State University.

Pablo Groisman (Buenos Aires): A Fleming-Viot process associated to sub-critical branching and a selection principle.

Let X be a continuous time Markov chain in a countable state space S with an absorbing state that we call 0. In this context it is natural to study the distribution of the process conditioned on non-absorption, which we call the conditioned evolution. A quasi-stationary distribution (qsd) is a probability measure that is invariant for the conditioned evolution. These distributions are important since they represent the state of the process at large times for a typical path that has not been absorbed.
The Fleming-Viot process associated to X (FV) is a particle system with state space S^N. At the beginning there are N (particles) independent copies of X, but when a particle is absorbed, jumps instantaneously over one of the N-1 particles that are not absorbed at that time. After that time, each particle evolves independently of the others.
The empirical measure of FV mimics the conditioned evolution of X. This fact makes FV a natural candidate to simulate the conditioned evolution for large times and also quasi-stationary distributions.
We will consider X to be a sub-critical branching process. In this case there is an infinite number of qsd, but there is a minimal one that has the smallest mean absorption time. We prove that (i) the empirical measure of FV converges to the conditioned evolution, (ii) FV is ergodic for every N, and (iii) the empirical measure of FV under its own invariant measure converges to the minimal qsd. That is, FV selects the minimal qsd. This is a joint work (in progress) with A. Asselah, P. Ferrari and M. Jonckheere.

Claude Le Bris (Marne la Vallée): Elements of mathematical formalization for some accelerated techniques in Molecular Dynamics.

We will present recent and ongoing works, in collaboration with Tony Lelievre and other colleagues, that all aim at mathematically formalizing various techniques introduced about a decade ago by A. Voter (Los Alamos) to accelerate molecular dynamics techniques. Exploring a physical energy landscape (configuration or phase space) using a molecular dynamics type approach is a challenging issue because the dynamics is a succession of long stays in metastable states and rapid transitions between them. Accelerating the dynamics is thus mandatory for the numerical practice in order to eventually visit all attainable states. It is a challenging issue per se. But an even more challenging question is to also recover, behind the artificially accelerated dynamics, the underlying physical dynamics, in terms of actual sequence of states visited and actual time needed to visit them. We will describe some approaches, and the first elements of mathematical formalizations of them. A number of open theoretical and numerical issues will be mentioned. Reference: A mathematical formalization of the parallel replica dynamics, C. Le Bris, T. Lelievre, M. Luskin, D. Perez, submitted to Monte-Carlo Methods and Applications, available at http://arxiv.org/abs/1105.4636.

Denis Villemonais (Palaiseau): Approximation of Markov processes conditioned on non-absorption. pdf

We will study the long term behavior of the distribution of Markov processes with absorption, and see that it describes and explains non-trivial behaviors, such as the mortality plateau in biology. However, in most cases, there are a priori no qualitative informations for the distribution of a conditioned Markov process, while these informations are of first importance in applications. In order to overcome this difficulty, we show in a great generality an approximation method for such distributions. During the talk, numerical illustrations of this approximation method will also be presented.