Self-Interacting Diffusions and their use for Simulation & Stochastic Persistence in Population Dynamics
 Project responsable Michel Benaim Abstract This research proposal is subdivided in two projects (denoted below A and B) that are natural continuations of the research conducted by the applicant and his collaborators under the preceding SNF grants. It will involve the participation of one post doctoral student (Charles-Edouard Bréhier, MSc ENS Lyon, PhD ENS Cachan) and two PhD students (Carl-Erik Gauthier, MSc Neuchâtel and Edouard Strickler, Msc ENS Cachan) supervised by the applicant. The thesis of C-E.~Gauthier is part of project A. It was initiated under the current funding (SNF grant 200020-149871/1) and should be achieved by the end of this new project. The post doctoral research program of C-E.~Bréhier is part of project A. The thesis of E.~Strickler is part to project C and will start under this new funding. {A. Self Interacting Diffusions}. This project is concerned by a class of stochastic differential equations on a manifold having the form $dX_t = dW_t(X_t) - \eps_t \nabla V_t(X_t) dt$ where $W_t(.)$ is a Brownian vector field, $V_t(x) = \int_0^t V(x,X_s) ds$ and $\eps_t = 1$ (unnormalized-interaction) or $\eps_t = 1/t$ (normalized-interaction). It consists of three parts. Part A1 started under the current SNF grant (200020-149871/1) with the thesis of C-E.~Gauthier. The work in collaboration with C-E.~Gauthier has already produced three research papers dealing with unnormalized-interaction in the {\em self-repelling} case where $V(\cdot,\cdot)$ is a Mercer Kernel that can be finitely diagonalized in an orthonormal basis of eigenfunctions of the Laplace operator. Under this new project C-E.~Gauthier will continue his investigation of self-repelling processes with more general Mercer kernels and address some conjectures concerning self-attracting processes. Parts A2 and A3 will investigate the use of normalized self-interacting diffusions for the simulation of quasi-stationary distributions and the sampling of complex probability distribution. Part A2 will be conducted in collaboration with Bertrand Cloez (Montpellier) and Fabien Panloup (Toulouse). Part A3 is the postdoctoral research program of C-E.~Bréhier. Visits and exchanges with researchers (S. Herrmann, Dijon; O. Raimond, Paris X, P. Tarres, Paris IX) experts on questions related to the topics of this project are scheduled . {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. A new PhD student, Edouard Strickler, will start his thesis under the supervision of the applicant. The aim of the thesis is to analyze a class of Markov processes that are obtained by random switching between ecological ordinary differential equations that are competitive. The thesis will lie at the interface of two fields that have been thoroughly investigated by the applicant and his co-authors under the preceding and current SNF grants: {\em stochastic persistence} and {\em piecewise deterministic markov processes}. Collaborations and exchanges with some researchers with whom the applicant has recently collaborated on these topics (including S. Schreiber (Davis), J. Hofbauer (Wien), C. Lobry (Nice) for the first one; F. Malrieu (Tours) and P-A Zitt (Marne la Vallée) for the second one) may be part of the project. Keywords Stochastic Persistence, Markov processes, Processes with reinforcement, Piecewise deterministic Markov processes, Population Dynamics, Persistence, Brownian Polymers Type of project Fundamental research project Research area Mathématiques Method of financing FNS - Encouragement de projets (Div. I-III) Status Completed Start of project 1-10-2015 End of project 30-9-2017 Overall budget 378'171.00 Contact Michel Benaim