Variable length Markov chains and dynamical sources

Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical source for which we explicitly build the mapping. On two examples, the ``comb'' and the ``bamboo blossom'', we find a necessary and sufficient condition for the existence and the unicity of a stationary probability measure for the VLMC. These two examples are detailed in order to provide the associated Dirichlet series as well as the generating functions of word occurrences.Comment: 45 pages, 15 figure

Comparison principles and applications to mathematical modelling of vegetal meta-communities

This article partakes of the PEGASE project the goal of which is a better understanding of the mechanisms explaining the behaviour of species living in a network of forest patches linked by ecological corridors (hedges for instance). Actually we plan to study the effect of the fragmentation of the habitat on biodiversity. A simple neutral model for the evolution of abundances in a vegetal metacommunity is introduced. Migration between the communities is explicitely modelized in a deterministic way, while the reproduction process is dealt with using Wright-Fisher models, independently within each community. The large population limit of the model is considered. The hydrodynamic limit of this split-step method is proved to be the solution of a partial differential equation with a deterministic part coming from the migration process and a diffusion part due to the Wright-Fisher process. Finally, the diversity of the metacommunity is adressed through one of its indicator, the mean extinction time of a species. At the limit, using classical comparison principles, the exchange process between the communities is proved to slow down extinction. This shows that the existence of corridors seems to be good for the biodiversity

Random series of functions and applications

We study the continuity properties of trajectories for some random series of functions $\sum a_kf(\alpha X_k(\omega))$ where $a_k$ is a complex sequence, $X_k$ a sequence of real independent random variables, $f$ is a real valued function with period one and summable Fourier coefficients. We obtain almost sure continuity results for these periodic or almost periodic series for a large class of functions, where the "almost sure" does not depend on the function

Conformal Measures for Multidimensional Piecewise Invertible Maps

Given a piecewise invertible map T : X ! X and a weight g : X ! ]0; 1[, a conformal measure is a probability measure on X such that, for all measurable A ae X with T : A ! TA invertible, (TA) = Z A 1 g d with a constant ? 0. Such a measure is an essential tool for the study of equilibrium states. Assuming that the topological pressure of the boundary is small, that log g has bounded distortion and an irreducibility condition, we build such a conformal measure. R'esum'e Etant donn'es une application inversible par morceaux T : X ! X et un poids g : X !]0; 1[, une mesure de probabilit'e sur X est dite conforme si, pour tout mesurable A ae X avec T jA injective, (TA) = Z A 1 g d pour un ? 0 constant. Une telle mesure est un outil essentiel pour l"etude des 'etats d"equilibre. Sous les hypoth`eses que la pression topologique du bord est petite, que log g est `a distorsion born'ee et qu'une condition d'irr'eductibilit'e est satisfaite, nous construisons une telle mesure. 1 Int..