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.