Self-Repelling diffusions, Piecewise deterministic Markov processes & Stochastic Persistence in Population Dynamics

Project responsable |
Michel Benaim |

Abstract |
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. |

Keywords |
Piecewise deterministic Markov processes, Persistence, Markov processes, Brownian Polymers, Processes with reinforcement, Population Dynamics, Game theory |

Type of project | Fundamental research project |

Research area | Mathématiques |

Method of financing | FNS - Encouragement de projets (Div. I-III) |

Status | Completed |

Start of project | 1-10-2013 |

End of project | 30-9-2015 |

Overall budget | 351'765.00 |

Contact | Michel Benaim |