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Jean Bérard

Publications and source records attributed to Jean Bérard.

18 recordsLinked to original sources
arXiv2025

The comonotone flow of a stochastically monotone Feller process on the real line

We show that any stochastically monotone Feller semigroup on R can be extended by a consistent family of order-preserving Feller semigroups on the successive powers of R. We exhibit a specific such family, which is uniquely characterized by a maximality property with respect to the super-modular order on Rn. A consequence is that, in this fairly general setting, there always exists a coupling between n c{à}dl{à}g versions of the underlying Markov process starting from n distinct initial positions, which do not cross one another.

math.PR↗
arXiv2025

Sharpness for monotone absorbing Interacting Particle Systems

We prove a sharpness result for the dynamics of finite-range Interacting Particle Systems (IPS) on $\{0,1\}^{\Z^d}$, which generalizes to a whole class of IPS the sharpness result for the phase transition of the contact process obtained by Bezuidenhout and Grimmett~\cite{BezuidenhoutGrimmett1991}. More precisely, starting from an IPS that is monotone, ergodic, and which admits the all-zero configuration as an absorbing state, we prove that there exists an arbitrarily small perturbation of the dynamics which leads to an \emph{exponentially} ergodic IPS. This also extends the sharpness result previously established for (discrete-time) probabilistic cellular automata in \cite{Har} to the continuous-time setting of IPS.

math.PR↗
arXiv2023

Hydrodynamic limit of N-branching Markov processes

We consider the behaviour of branching-selection particle systems in the large population limit. The dynamics of these systems is the combination of the following three components: (a) Motion: particles move on the real line according to a continuous-time Markov process; (b) Branching: at rate 1, each particle gives birth to a new particle at its current location; (c) Selection: to keep the total number of particles constant, each branching event causes the particle currently located at the lowest position in the system to be removed instantly. Starting with N $\ge$ 1 particles whose positions at time t = 0 form an i.i.d. sample with distribution $μ$ 0 , we investigate the behaviour of the system at a further time t > 0, in the limit N $\rightarrow$ +$\infty$. Our first main result is that, under suitable (but rather mild) regularity assumptions on the underlying Markov process, the empirical distribution of the population of particles at time t converges to a deterministic limit, characterized as the distribution of the Markov process at time t conditional upon not crossing a certain (deterministic) moving boundary up to time t. Our second result is that, under additional regularity assumptions, the lowest particle position at time t converges to the moving boundary. These results extend and refine previous works done by other authors, that dealt mainly with the case where particles move according to a Brownian motion. For instance, our results hold for a wide class of L{é}vy processes and diffusion processes. Moreover, we obtain improved non-asymptotic bounds on the convergence speed.

math.PR↗
arXiv2022

A probabilistic cellular automaton that admits no successful basic i.i.d. coupling

In this paper, we revisit a classic example of probabilistic cellular automaton (PCA) on {0, 1} Z , namely, addition modulo 2 of the states of the left-and right-neighbouring cells, followed by either preserving the result of the addition, with probability p, or flipping it, with probability 1 -- p. It is well-known that, for any value of p $\in$]0, 1[, this PCA is ergodic. We show that, for p sufficiently close to 1, no coupling of the PCA dynamics based on the composition of i.i.d. random functions of nearest-neighbour states (we call this a basic i.i.d. coupling), can be successful, where successful means that, for any given cell, the probability that every possible initial condition leads to the same state after t time steps, goes to 1 as t goes to infinity. In particular, this precludes the possibility of a CFTP scheme being based on such a coupling. This property stands in sharp contrast with the case of monotone PCA, for which, as soon as ergodicity holds, there exists a successful basic i.i.d. coupling.

math.PR↗
arXiv2019

A Coupling Proof of Convex Ordering for Compound Distributions

In this paper, we give an alternative proof of the fact that, when compounding a nonnegative probability distribution, convex ordering between the distributions of the number of summands implies convex ordering between the resulting compound distributions. Although this is a classical textbook result in risk theory, our proof exhibits a concrete coupling between the compound distributions being compared, using the representation of one-period discrete martingale laws as a mixture of the corresponding extremal measures.

math.PR↗
arXiv2016

Fluctuations of the front in a one-dimensional model for the spread of an infection

We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent nearest-neighbor continuous-time symmetric random walks on the integer lattice $\mathbb{Z}$ with jump rates $D_R$ for red particles and $D_B$ for blue particles, the interaction rule being that blue particles turn red upon contact with a red particle. The initial condition consists of i.i.d. Poisson particle numbers at each site, with particles at the left of the origin being red, while particles at the right of the origin are blue. We are interested in the dynamics of the front, defined as the rightmost position of a red particle. For the case $D_R=D_B$, Kesten and Sidoravicius established that the front moves ballistically, and more precisely that it satisfies a law of large numbers. Their proof is based on a multi-scale renormalization technique, combined with approximate sub-additivity arguments. In this paper, we build a renewal structure for the front propagation process, and as a corollary we obtain a central limit theorem for the front when $D_R=D_B$. Moreover, this result can be extended to the case where $D_R>D_B$, up to modifying the dynamics so that blue particles turn red upon contact with a site that has previously been occupied by a red particle. Our approach extends the renewal structure approach developed by Comets, Quastel and Ram\'{ı}rez for the so-called frog model, which corresponds to the $D_B=0$ case.

math.PR↗
arXiv2013

The limiting process of $N$-particle branching random walk with polynomial tails

We consider a system of $N$ particles on the real line that evolves through iteration of the following steps: 1) every particle splits into two, 2) each particle jumps according to a prescribed displacement distribution supported on the positive reals and 3) only the $N$ right-most particles are retained, the others being removed from the system. This system has been introduced in the physics literature as an example of a microscopic stochastic model describing the propagation of a front. Its behavior for large $N$ is now well understood -- both from a physical and mathematical viewpoint -- in the case where the displacement distribution admits exponential moments. Here, we consider the case of displacements with regularly varying tails, where the relevant space and time scales are markedly different. We characterize the behavior of the system for two distinct asymptotic regimes. First, we prove convergence in law of the rescaled positions of the particles on a time scale of order $\log N$ and give a construction of the limit based on the records of a space-time Poisson point process. Second, we determine the appropriate scaling when we let first the time horizon, then $N$ go to infinity.

math.PR↗
arXiv2013

A Lognormal Central Limit Theorem for Particle Approximations of Normalizing Constants

This paper deals with the numerical approximation of normalizing constants produced by particle methods, in the general framework of Feynman-Kac sequences of measures. It is well-known that the corresponding estimates satisfy a central limit theorem for a fixed time horizon $n$ as the number of particles $N$ goes to infinity. Here, we study the situation where both $n$ and $N$ go to infinity in such a way that $\lim_{n\rightarrow\infty}% n/N=α>0$. In this context, Pitt et al. \cite{pitt2012} recently conjectured that a lognormal central limit theorem should hold. We formally establish this result here, under general regularity assumptions on the model. We also discuss special classes of models (time-homogeneous environment and ergodic random environment) for which more explicit descriptions of the limiting bias and variance can be obtained.

math.PR↗
arXiv2012

Coupling from the past times with ambiguities and perturbations of interacting particle systems

We discuss coupling from the past techniques (CFTP) for perturbations of interacting particle systems on the d-dimensional integer lattice, with a finite set of states, within the framework of the graphical construction of the dynamics based on Poisson processes. We first develop general results for what we call CFTP times with ambiguities. These are analogous to classical coupling (from the past) times, except that the coupling property holds only provided that some ambiguities concerning the stochastic evolution of the system are resolved. If these ambiguities are rare enough on average, CFTP times with ambiguities can be used to build actual CFTP times, whose properties can be controlled in terms of those of the original CFTP time with ambiguities. We then prove a general perturbation result, which can be stated informally as follows. Start with an interacting particle system possessing a CFTP time whose definition involves the exploration of an exponentially integrable number of points in the graphical construction, and which satisfies the positive rates property. Then consider a perturbation obtained by adding new transitions to the original dynamics. Our result states that, provided that the perturbation is small enough (in the sense of small enough rates), the perturbed interacting particle system too possesses a CFTP time (with nice properties such as an exponentially decaying tail). The proof consists in defining a CFTP time with ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed dynamics. Finally, we discuss examples of particle systems to which this result can be applied. Concrete examples include a class of neighbor-dependent nucleotide substitution model, and variations of the classical voter model, illustrating the ability of our approach to go beyond the case of weakly interacting particle systems.

math.PR↗
arXiv2010

Survival probability of the branching random walk killed below a linear boundary

We give an alternative proof of a result by N. Gantert, Y. Hu and Z. Shi on the asymptotic behavior of the survival probability of the branching random walk killed below a linear boundary, in the special case of deterministic binary branching and bounded random walk steps. Connections with the Brunet-Derrida theory of stochastic fronts are discussed.

math.PR↗
arXiv2010

Brunet-Derrida behavior of branching-selection particle systems on the line

We consider a class of branching-selection particle systems on $\R$ similar to the one considered by E. Brunet and B. Derrida in their 1997 paper "Shift in the velocity of a front due to a cutoff". Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size $N$ of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate $(\log N)^{-2}$. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of $N$ independent branching random walks killed below a linear space-time barrier.

math.PR↗
arXiv2008

An example of Brunet-Derrida behavior for a branching-selection particle system on $\Z$

We consider a branching-selection particle system on $\Z$ with $N \geq 1$ particles. During a branching step, each particle is replaced by two new particles, whose positions are shifted from that of the original particle by independently performing two random walk steps according to the distribution $p δ_{1} + (1-p) δ_{0}$, from the location of the original particle. During the selection step that follows, only the N rightmost particles are kept among the 2N particles obtained at the branching step, to form a new population of $N$ particles. After a large number of iterated branching-selection steps, the displacement of the whole population of $N$ particles is ballistic, with deterministic asymptotic speed $v_{N}(p)$. As $N$ goes to infinity, $v_{N}(p)$ converges to a finite limit $v_{\infty}(p)$. The main result is that, for every $0<p<1/2$, as $N$ goes to infinity, the order of magnitude of the difference $v_{\infty}(p)- v_{N}(p)$ is $\log(N)^{-2}$. This is called Brunet-Derrida behavior in reference to the 1997 paper by E. Brunet and B. Derrida "Shift in the velocity of a front due to a cutoff" (see the reference within the paper), where such a behavior is established for a similar branching-selection particle system, using both numerical simulations and heuristic arguments.

math.PR↗
arXiv2008

Large Deviations of the Front in a one dimensional model of $X+Y \to 2X$

We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction $X+Y \to 2X$ on the integer lattice in which $Y$ particles do not move while $X$ particles move as independent simple continuous time random walks of total jump rate $2$. For a wide class of initial conditions, we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval $[0,v]$, where $v$ is the velocity of the front given by the law of large numbers. We also give more precise estimates for the rate of decay of the slowdown probabilities. Our results indicate a gapless property of the generator of the process as seen from the front, as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state.

math.PR↗
arXiv2007

Coupling times with ambiguities for particle systems and applications to context-dependent DNA substitution models

We define a notion of coupling time with ambiguities for interacting particle systems, and show how this can be used to prove ergodicity and to bound the convergence time to equilibrium and the decay of correlations at equilibrium. A motivation is to provide simple conditions which ensure that perturbed particle systems share some properties of the underlying unperturbed system. We apply these results to context-dependent substitution models recently introduced by molecular biologists as descriptions of DNA evolution processes. These models take into account the influence of the neighboring bases on the substitution probabilities at a site of the DNA sequence, as opposed to most usual substitution models which assume that sites evolve independently of each other.

math.PR↗
arXiv2005

Solvable models of neighbor-dependent nucleotide substitution processes

We prove that a wide class of models of Markov neighbor-dependent substitution processes on the integer line is solvable. This class contains some models of nucleotide substitutions recently introduced and studied empirically by molecular biologists. We show that the polynucleotide frequencies at equilibrium solve explicit finite-size linear systems. Finally, the dynamics of the process and the distribution at equilibrium exhibit some stringent, rather unexpected, independence properties. For example, nucleotide sites at distance at least three evolve independently, and the sites, if encoded as purines and pyrimidines, evolve independently.

math.PR↗
arXiv2004

Asymptotics of a two-dimensional sticky random walk

We study the asymptotic behavior of a Markov chain on $\mathbb{Z}^2$ that corresponds to the two-dimensional marginals of a reinforcement process on $\mathbb{Z}^{\mathbb{N}}$. Three distinct asymptotic regimes are identified, depending on the scaling of the reinforcement parameter $Δ$ with respect to the number of steps performed by the chain.

math.PR↗