Stochastic Approximations and Applications & Stochastic Persistence in Population Dynamics
| Responsable du projet | Michel Benaim |
| Collaborateur |
Yoann Offret
Bastien Marmet Basile De Loynes |
| Résumé |
This research proposal is subdivided in three projects (denoted
below A, B and C) and a postdoctoral research project presenting
challenging connections with project A. Projects A and B are a
natural continuation of the research initiated and conducted by the
applicant and his collaborators under the preceding SNF grants and C
is a new project. {A. Stochastic Approximations and applications}
Project A deals with the applications of the dynamical system
approach to stochastic approximation - developed by the applicant
and his collaborators before and under the preceding SNF grants -
to certain questions arising in the theory of learning in games. -
In collaboration with M. Faure (currently in Neuchâtel) and
O. Raimond (Paris 10) we will analyze the consistency and
efficiency of a class of adaptive learning rules including {\em
vanishing smooth fictitious play and Markovian fictitious play.} -
In collaboration with J. Hofbauer (Vienna) and S. Sorin (Paris 6)
we will consider stochastic approximation with constant step sizes
that are associated to a differential inclusion. This line of
research is motivated by several population models arising
especially in evolutionary game theory. {Post doctoral research
project.} The post-doctoral student involved in this part of the
program is Erwan Hillion, a former PhD student of M. Ledoux, who
obtained his PhD in June 2010. The main research interest of Erwan
Hillion is the theory of Ricci curvature bounds on metric-measure
spaces, recently developed by Lott, Villani and Sturm and the
generalization of Lott-Villani-Sturm theory to discrete spaces. In
connection with project A, Erwan Hillion will explore how certain
ideas and technics from optimal transportation can be applied for
analyzing the behavior of learning processes associated to {\em
potential games}. The key idea is that the behavior of such
processes can be characterized by a certain functional which is
formally a McKean Vlassov non linear free energy defined on a the
space of probabilities over a {\em finite} set. A challenging
question is to see if Bakry-Emery type conditions that ensure {\em
displacement convexity} of this functional when the base space is a
Riemannian manifold can be generalized to the situation where the
base space is a finite set. {B. Persistence for randomly perturbed
systems.} This project is concerned with the question of
persistence for ecological models of species in interaction
subjected to random fluctuations. It is a continuation of the work
initiated under the precedings SNF grants in collaboration with J.
Hofbauer (Vienna), B. Sandholm (Madison) and S. Schreiber (UC
Davis). It will be conducted in close collaboration with
S.~Schreiber and possibly J.~Hofbauer (University of Vienna,
Austria) and, on certain aspects, E.~Locherbach (University
Cergy-Pontoise, France). - The main goal is to pursue our analysis
of general conditions ensuring the persistence (i.e non extinction)
of stochastic models of interacting species {\bf beyond} small
perturbations of deterministic systems. - A new PhD thesis, related
to this project will be supervised by the applicant. The PhD
student, Bastien Marmet, will investigate the connections between
stochastic persistence and quasi-stationary distributions for
certain population processes. {C. Random composition of
deterministic flows.} This last part is a new research project
dealing with the qualitative and quantitative analysis of a class
of Markov processes that are obtained as a random composition of
deterministic flows integrated over random times. The motivation
come from the analysis of certain processes arising in molecular
biology. This part of the program will involve the collaborations
of F. Malrieu and S. LeBorgne (Université de Rennes, France)
and P-A. Zitt (Université de Bourgogne, France). - Our first
goal is to investigate qualitative properties of these processes
(existence of invariant measures, nature of the support, ergodicity
and convergence in total variation of the law). - Our second and
main goal is to provide quantitative bounds in term of the
Wasserstein distance of the rate at which -under certain
assumptions- the process converges in law to its invariant
measure. |
| Mots-clés |
Stochastic processes, Stochastic approximation, Game Theory, Consistency, Permanence, Quasistationnary distribution, random composition of flows |
| Type de projet | Recherche fondamentale |
| Domaine de recherche | Mathématiques |
| Source de financement | FNS - Encouragement de projets (Div. I-III) |
| Etat | Terminé |
| Début de projet | 1-10-2011 |
| Fin du projet | 30-9-2013 |
| Budget alloué | 355'845.00 |
| Autre information |
http://p3.snf.ch/projects-138242# |
| Contact | Michel Benaim |