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Research interests

My current research interests are in

Probability theory applied to evolutionary biology and ecology

This part of my work contains joint works with F. Campillo, R. Ferrière, C. Fritsch, B. Henry, P.-E. Jabin, A. Lambert, S. Méléard, M. Richard and S. Roelly.

Biological framework

I am interested in the study, the simulation and the microscopic and macroscopic modeling of populations subject to evolution, from the viewpoint of ecology (interactions in the population are precisely modeled), in the biological theoretical framework of adaptive dynamics, of population genetics, and more generally in population dynamics.

One of the problem I am studying is the mathematical understanding of the phenomenon of evolutionary branching, where a population initially concentrated around a single phenotype is driven by the evolutionary forces to states where two (or more) distinct phenotypes coexist, without geographical separation. This phenomenon is at the origin of diversity in ecosystems, and is believed to be an important mechanism for sympatric speciation, i.e. separation of a single species into two without geographical separation. I have developed and participated in two approaches to this question, the first one based on stochastic individual-based models, and the second one based on the asymptotic analysis of nonlocal PDEs.

I am also interested in the study of spatial organisation (clustering) of spatially structured populations, of robustness and of evolution by punctuated equilibria. In this context, I also study the spatial structuration of clonal plants via stochastic individual-based models. Another line of research I am currently developing in collaboration with C. Fritsch during her post-doc concerns the growth of bacterial populations structured by mass within a chemostat.

I am also interested in the population genetics of branching populations, where I try to extend classical properties of constant population size models (Wright-Fisher, Cannings or Moran models) to populations growing according to general branching processes. In particular, I am studying the frequency spectrum in such a population subject to neutral mutations.

Finally, stochastic models of growth of tumoral populations taking into account their pH regulation mechanisms were developed in collaboration with J. Pouyssegur, N. Mazure, D. Talay during the PhD thesis of J. Claisse.

Mathematical framework

The basic natural models in this biological setting are measure-valued interacting particles systems with birth, death, competition, mutation, resource consumption, dispersal or spatial diffusion.

I am interested in various biologistically realistic scalings of this microscopic model, yielding the convergence to, on the one hand, PDEs, SPDEs, deterministic or stochastic integro-differential equations and superprocesses; and, on the other hand, to jump processes, ODEs or diffusion processes, by multi-scale analysis (timescales separation). I am also interested in probabilistic interpretations of some other natural PDE models arising in this setting, that can be used to study evolutionary branching from the analytical side.

I also study the long time behaviour of these limiting models, linked to the biological phenomenon of evolutionary branching. In the case of the diffusion processes, this problem is linked to the large deviations theory and to the problem of exit from a domain. The mathematical study of evolutionary branching also requires fine properties of long time behaviour of competitive Lotka-Volterra systems.

In population genetics, out work on branching populations relies strongly on special properties of appropriate contour processes, which allow to recover a Markovian property (Lévy process) in models where life-lengths need not to be exponential. This is the subject of the current PhD thesis of B. Henry.

Finally, an important part of my work is devoted to numerical simulations of the microscopic models and their scaling limits, specifically for spatially structured populations.

Probability theory applied to molecular dynamics

This part of my work is developed in three main directions:

  • The probabilistic interpretations of divergence form operators with discontinuous coefficients in any dimension, with applications to the numerical resolution of the Poisson-Boltzmann equation solved by the electrostatic potential around a given spatial configuration of a biolecular assembly. The probabilistic interpretation of such operators makes use of the local time of auxiliary processes. This interpretation allows us to propose various probabilistic resolution algorithms for the Poisson-Boltzmann equation. A recent extension to the non-linear Poisson-Boltzmann equation has also been made, based of branching diffusions. A work is also currently being finished on the probabilistic interpretation of the non-linear Poisson-Boltzmann equation using backward stochastic differential equations. This is joint work with M. Bossy, H. Leman, S. Maire, N. Perrin, D. Talay, L.Violeau and M. Yvinec.
  • The Fourier analysis of covariance matrices with delay obtained from a time series of the dynamcis of a biomolecular assembly, in order to identify fast and slow components and to construct simplified stochastic dynamics. Preliminary resuls on this work were obtained in the PhD thesis of N. Perrin under the cosupervision of D. Talay and myself.
  • The analysis of new statistical techniques for solving very ill-posed least square problems arising from molecular dynamics applications. This is joint work with C. Chipot and E. Faou.

Probabilistic interpretation of PDEs

This point is developed mainly for the Poisson-Boltzmann equation mentioned above as an application to molecular dynamics. My interest lies in the theoretical development of probabilistic interpretations of linear and non-linear equations with stochastic differential equations, branching diffusions and backward stochastic differential equations, and in the practical implementation of these methods. Apart from the applications to Poisson-Boltzmann equation, I am currently working on the use of branching diffusions and large deviations to obtain a new interpretation of the Hamilton-Jacobi equation with constraint arising from the PDE approach to adaptive dynamics and evolutionary branching. I am also working on variance reduction techniques to solve some of the drawbacks of Feynman-Kac formulas based on branching diffusions.

Quasi-stationary distributions

For general absorbed Markov processes, quasi-stationary distributions (QSDs) are distributions which are invariant for the process conditionally on non-absorption. I developed recently with D. Villemonais general necessary and sufficient criteria to ensure the existence, uniqueness, and exponential convergence w.r.t. the total variation norm to QSDs. These results allow us to extend and generalize many results obtained in the past years to a large variety of processes (birth and deat processes with catastrophes, neutron transport processes, individual-based models). We are currently working on the implications of our results on one-dimensional diffusion with killing, and to multi-dimensional diffusions with K. Coulibaly-Pasquier. My works with P. Diaconis and L. Miclo and with S. Roelly also concern this topic.

Stochastic differential equations

SDE are involved in many parts of my work as a tool for modeling of to solve PDEs using Monte-Carlo techniques. I also worked on strong existence and pathwise uniqueness for SDEs with non-smooth coefficients in collaboration with P.-E. Jabin. One of the extensions of these results I am currently working on concerns the construction and the strong error analysis of Euler schemes for SDEs with non-smooth coefficients.

Dynamical systems

This point was already mentionned above concerning the mathematical study of evolutionary branching (see my work with P.-E. Jabin and G. Raoul). I am also interested in the question of well-posedness for dynamical systems. Jointly with P.-E. Jabin, we improved the existing conditions on the force fields in Hamiltonian dynamical systems to ensure existence and uniqueness when the force field has less than one derivative (H^3/4).

Stochastic control

My interest in this area comes from three applications:

  • Within an industrial collaboration with the SME Alphability, TOSCA Nancy is working on the tail estimation of asset returns and on the computation of risk indicators such as the well-known Value-at-Risk. One of our goal is to develop portfolio optmization tools taking into account risk constraints.
  • I am studying jointly with S. Maroso, E. Tanre and D. Talay stochastic control problems related to the hedging of barrier options. A first part of the work is devoted to the problem of super-replication of such assets under Gamma constraints. A second part is devoted to the mathematical and numerical analysis of various hedging strategies where only finitely many portfolio reallocations are allowed with delay constrints.
  • The PhD thesis of J. Claisse, cosupervised by D. Talay and myself, was devoted to stochastic control problems for population dynamics. We are currently working on the links between ergodic control of population dynamics and quasi-stationary distributions.

Main mathematical tools

  • Measure-valued interacting particles systems (birth, death, competition, mutation, dispersal processes)
  • General branching processes (splitting trees)
  • General birth and death processes
  • Multi-scale analysis
  • Large deviations and problem of exit from a domain
  • Stochastic differential equations
  • Martingale problems
  • Probabilistic interpretations of PDEs
  • Quasi-stationary distributions
  • Dynamical systems of Lotka-Volterra type, of chemostat type or of Hamiltonian type
  • Dawson-Watanabe superprocesses
  • Fourier analysis
  • Spectral analysis
  • Coupling techniques
  • Stochastic control

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