This research proposal is subdivided in three projects (denoted below A, B, C) that are natural continuations of the research conducted by the applicant and his collaborators under the preceding SNF grants and a post doctoral research project (project D). It will involve the participation of a post doctoral students (M. Oliu-Barton) and two PhD students (B. Barmet and C. E. Gauthier) supervised by the applicant. The thesis of C. E. Gauthier is related to project A and will start under this new funding. The thesis of B. Marmet -related to project B- was initiated under the preceding funding (SNF grant 138242) and should be achieved by the end of the academic year 2013/2014. The post doctoral research program of B.~Cloez is part of project C. The post doctoral research program of M.~ Oliu-Barton is strongly connected to the work of the applicant in game theory and stochastic approximations (see the SNF grants 138242, 112316, 103625) - {A. Self Repelling Diffusions} The main purpose of this project is to analyze the long term behavior of a class of stochastic differential equations on a compact manifold having the form $dX_t = dW_t(X_t) - \nabla V_t(X_t)$ where $W_t(.)$ is a Brownian vector field, $V_t(x) = \int_0^t V(x,X_s)$ and $V(\cdot,\cdot)$ is a Mercer Kernel. Such processes have been used in the literature as models of Brownian polymers and self-repelling processes. Our main goal is to provide general conditions ensuring that the process converge in distribution toward a product measure and to estimate its rate of convergence. Visits of O. Raimond (Paris X) and B. Schapira (Marseille) are scheduled to work on this project. A new PhD thesis, related to this project, will be supervised by the applicant. The PhD student, Carl-Erik Gauthier, will investigate the long term behavior of self-repelling processes on "simple" manifolds (such as the torus or the $n$ sphere) for particular Mercer kernel which can be finitely diagonalized in an orthonormal basis consisting of eigenfunctions of the Laplace operator. {B. Stochastic Persistence in Population Dynamics} 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 preceding SNF grants (112316, 120218, 130574) in collaboration with J. Hofbauer (Vienna), B. Sandholm (Madison) and S. Schreiber (UC Davis). A thesis, supervised by the applicant, started in 2011 under the preceding grant (138242). The PhD student Bastien Marmet has analyzed the limiting behavior of quasi-stationary distributions for finite population models (in the limit of infinite populations) under the assumption that a certain deterministic system (obtained by mean field approximation) is persistent (i.e admits an global attractor bounded away from the extinction set). In this second phase of the thesis he will consider similar problems for certain classes of stochastic differential equations that can be obtained as diffusion approximations of populations models. In collaboration with S.~Schreiber (UC davis), possibly J.~Hofbauer (University of Vienna, Austria) and, on certain aspects, E.~Locherbach (University Cergy-Pontoise, France), we will 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. {C. Piecewise deterministic Markov processes} This last project is the continuation of a research project initiated under the preceding SNF grant. It deals 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 Marne la Vallée, France) We will pursue our investigation of the qualitative properties of these processes (existence of invariant measures, nature of the support, ergodicity and convergence in total variation of the law) with a special emphasis on non compact state spaces.