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 |