P. Collet

We develop a transport-entropy framework for Gaussian concentration inequalities on the infinite product space $S^{\mathbb Z^d}$, where $S$ is a finite set, in which sensitivity is measured by the $\ell^2$-norm of local oscillations. We…

We consider a general class of birth-and-death processes with state space $\{0,1,2,3,\ldots\}$ which describes the size of a population going eventually to extinction with probability one. We obtain the complete spectrum of the generator of…

We consider spin-flip dynamics of configurations in $\{-1,1\}^{\mathbb{Z}^d}$, and study the time evolution of concentration inequalities. For "weakly interacting" dynamics we show that the Gaussian concentration bound is conserved in the…

We consider the behavior of the Gaussian concentration bound (GCB) under stochastic time evolution. More precisely, we consider a Markovian diffusion process on $\mathbb{R}^d$ and start the process from an initial distribution $\mu$ that…

We study a class of multi-species birth-and-death processes going almost surely to extinction and admitting a unique quasi-stationary distribution (qsd for short). When rescaled by $K$ and in the limit $K\to+\infty$, the realizations of…

We consider a class of birth-and-death processes describing a population made of $d$ sub-populations of different types which interact with one another. The state space is $\mathbb{Z}_+^d$ (unbounded). We assume that the population goes…

We consider Gibbs measures on the configuration space $S^{\mathbb{Z}^d}$, where mostly $d\geq 2$ and $S$ is a finite set. We start by a short review on concentration inequalities for Gibbs measures. In the Dobrushin uniqueness regime, we…

We study a general class of birth-and-death processes with state space $\mathbb{N}$ that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is…

We consider the problem of reconstructing a function given its values on a set of points with finite density. We prove that with probability one, the values of an almost periodic function on a random array of points (with finite density)…

We consider the initial value problem for the thermal-diffusive combustion systems of the form: $u_{1,t}= Delta_{x}u_1 - u_1 u_2^m$, $u_{2,t}= d Delta_{x} u_2 + u_1 u_2^m$, $x in R^{n}$, $n geq 1$, $m geq 1$, $d > 1$, with bounded uniformly…

We consider a restricted Solid-on-Solid interface in $\Bbb{Z}_{+}$, subject to a potential $V\left( n\right) $ behaving at infinity like $-\mathrm{w}/n^{2}$. Whenever there is a wetting transition as $b_{0}\equiv \exp V\left( 0\right) $ is…

We consider the development of instabilities of homogeneous stationary solutions of discrete time lattice maps. Under some generic hypothesis we derive an amplitude equation which is the space-time continuous Ginzburg-Landau equation. Using…

We study the number of visits to balls B_r(x), up to time t/mu(B_r(x)), for a class of non-uniformly hyperbolic dynamical systems, where mu is the SRB measure. Outside a set of `bad' centers x, we prove that this number is approximately…

In this paper, we apply Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance…

In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps…

We define the topological entropy per unit volume in parabolic PDE's such as the complex Ginzburg-Landau equation, and show that it exists, and is bounded by the upper Hausdorff dimension times the maximal expansion rate. We then give a…

We investigate the existence of a global semiflow for the complex Ginzburg-Landau equation on the space of bounded functions in unbounded domain. This semiflow is proven to exist in dimension 1 and 2 for any parameter values of the standard…

We consider matching with shifts for Gibbsian sequences. We prove that the maximal overlap behaves as $c\log n$, where $c$ is explicitly identified in terms of the thermodynamic quantities (pressure) of the underlying potential. Our…

For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^n\to\R$ which are componentwise Lipschitz. The proof is based on coupling and decay of…

For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost sure central limit theorem (ASCLT). In fact, we provide a speed of convergence in the Kantorovich metric. Maxima of…