Long-range exclusion processes, generator and invariant measures

We show that if $\mu$ is an invariant measure for the long range exclusion process putting no mass on the full configuration, $L$ is the formal generator of that process and $f$ is a cylinder function, then $Lf\in\mathbf{L}^1(d\mu)$ and $\int Lf d\mu=0$. This result is then applied to determine (i) the set of invariant and translation-invariant measures of the long range exclusion process on $\mathbb{Z}^d$ when the underlying random walk is irreducible; (ii) the set of invariant measures of the long range exclusion process on $\mathbb{Z}$ when the underlying random walk is irreducible and either has zero mean or allows jumps only to the nearest-neighbors.Comment: Published at http://dx.doi.org/10.1214/009117905000000486 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Euler hydrodynamics for attractive particle systems in random environment

We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on $\Z$ in random ergodic environment. Our result is a strong law of large numbers, that we illustrate with various examples.Comment: 36 page

About the long range exclusion process

International audienceIntroduced by Spitzer [23] and studied by Liggett [14] the Long Range Exclusion Process (LREP) is an interacting particle system with truly long range interaction. Informally speaking: each particle on a lattice hops at independent random times following instantaneously a random dynamic on the lattice until finding a vacant site (if any). These instantaneous, potentially long jumps prevent the process to have the Feller property. In this paper we review the main results about the LREP including recent developments obtained in [11,24] and [4]. New results on Feller approximations and about the regularity set of the LREP are also provided. Finally we briefly discuss some connections of the LREP with the discrete Hammersley process introduced in [8] and the sandpile process in infinite volume developed in [18] and [17]

A stochastic model of evolution

National audienceWe propose a stochastic model for evolution. Births and deaths of species occur with constant probabilities

About the long range exclusion process

International audienceIntroduced by Spitzer [23] and studied by Liggett [14] the Long Range Exclusion Process (LREP) is an interacting particle system with truly long range interaction. Informally speaking: each particle on a lattice hops at independent random times following instantaneously a random dynamic on the lattice until finding a vacant site (if any). These instantaneous, potentially long jumps prevent the process to have the Feller property. In this paper we review the main results about the LREP including recent developments obtained in [11,24] and [4]. New results on Feller approximations and about the regularity set of the LREP are also provided. Finally we briefly discuss some connections of the LREP with the discrete Hammersley process introduced in [8] and the sandpile process in infinite volume developed in [18] and [17]

Some properties of k-step exclusion processes

International audienceWe introduce k-step exclusion processes as generalizations of the simple exclusion process. We state their main equilibrium properties when the underlying stochastic matrix corresponds to a random walk or is positive recurrent and reversible. Finally, we prove laws of large numbers for tagged and second-class particles

Un résultat pour le processus d'exclusion à longue portée.

International audienceFor long range exclusion processes we show, under the hypothesis of irreducibility and translation invariance of the underlying transition matrix, that all invariant and translation invariant measures are mixture of Bernoulli product measures with constant density

Questions for second class particles in exclusion processes.

International audienc

Relaxation time to equilibrium of the one-dimensional symmetric zero range process with constant rate

International audienceWe prove that the one-dimensional symmetric zero range dynamics, starting either with a periodic configuration or with a stationary exponential mixing probability distribution, converges to equilibrium faster than log t divided by square root of t