Publications

2013

• Quantifying the Mutational Meltdown in Diploid Populations
  
  • Coron Camille
  • Méléard Sylvie
  • Porcher Emmanuelle
  • Robert Alexandre
  The American Naturalist, University of Chicago Press, 2013, 181 (5), pp.623-636. Mutational meltdown, in which demographic and genetic processes mutually reinforce one another to accelerate the extinction of small populations, has been poorly quantified despite its potential importance in conservation biology. Here we present a model-based framework to study and quantify the mutational meltdown in a finite diploid population that is evolving continuously in time and subject to resource competition. We model slightly deleterious mutations affecting the population demographic parameters and study how the rate of mutation fixation increases as the genetic load increases, a process that we investigate at two timescales: an ecological scale and a mutational scale. Unlike most previous studies, we treat population size as a random process in continuous time. We show that as deleterious mutations accumulate, the decrease in mean population size accelerates with time relative to a null model with a constant mean fixation time. We quantify this mutational meltdown via the change in the mean fixation time after each new mutation fixation, and we show that the meltdown appears less severe than predicted by earlier theoretical work. We also emphasize that mean population size alone can be a misleading index of the risk of population extinction, which could be better evaluated with additional information on demographic parameters. (10.1086/670022)
  DOI : 10.1086/670022
• Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations
  
  • Amirat Youcef
  • Hamdache Kamel
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2013, 33 (8), pp.3289-3320. In this paper we study the equations of flow and heat transfer in a magnetic fluid with internal rotations, when the fluid is subjected to the action of an external magnetic field. The system of equations is a combination of the Navier-Stokes equations, the magnetization relaxation equation of Bloch type, the magnetostatic equations and the temperature equation. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions and establish a blow-up criterium for strong solutions. We then prove the global-in-time existence of strong solutions, under smallness assumptions on the initial data and the external magnetic field. (10.3934/dcds.2013.33.3289)
  DOI : 10.3934/dcds.2013.33.3289
• Some limit theorems for Hawkes processes and application to financial statistics
  
  • Bacry Emmanuel
  • Delattre Sylvain
  • Hoffmann Marc
  • Muzy Jean-François
  Stochastic Processes and their Applications, Elsevier, 2013, 123 (7), pp.2475 - 2499. Abstract In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [ 0 , T ] when T ? ? . We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead?lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms. (10.1016/j.spa.2013.04.007)
  DOI : 10.1016/j.spa.2013.04.007
• Log-normal continuous cascade model of asset returns: aggregation properties and estimation
  
  • Bacry Emmanuel
  • Kozhemyak Alexey
  • Muzy Jean-François
  Quantitative Finance, Taylor & Francis (Routledge), 2013, 13 (5), pp.795-818. no abstract (10.1080/14697688.2011.647411)
  DOI : 10.1080/14697688.2011.647411
• The 2d-Directed Spanning Forest is almost surely a tree
  
  • Coupier David
  • Tran Viet Chi
  Random Structures and Algorithms, Wiley, 2013, 42 (1), pp.59-72. We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi-infinite paths in the DSF. (10.1002/rsa.20400)
  DOI : 10.1002/rsa.20400
• A geometric approach for convexity in some variational problem in the Gauss space
  
  • Goldman Michael
  Rendiconti del Seminario Matematico della Università di Padova, University of Padua / European Mathematical Society, 2013. In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.