No abstract

International audienceWe derive by a constructive method the hydrodynamic behavior of attractive processes with irreducible jumps and product invariant measures. Our approach relies on (i) explicit construction of Riemann solutions without assuming convexity, which may lead to contact discontinuities and (ii) a general result which proves that the hydrodynamic limit for Riemann initial profiles implies the same for general initial pro5les. The k-step exclusion process provides a simple example. We also give a law of large numbers for the tagged particle in a nearest neighbor asymmetric k-step exclusion process

We prove almost sure Euler hydrodynamics for a large class of attractive particle systems on Z starting from an arbitrary initial profile. We generalize earlier works by Seppäläinen (1999) and Andjel et al. (2004). Our constructive approach requires new ideas since the subadditive ergodic theorem (central to previous works) is no longer effective in our setting. 0 AMS 2000 subject classification. Primary 60K35; Secondary 82C22.

We consider attractive irreducible conservative particle systems on $\mathbb{Z}$, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1--30], from which we relax the assumption that the process has explicit invariant measures.Comment: Published at http://dx.doi.org/10.1214/009117906000000115 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality 1 . 1 MSC 2000: Primary-82C22; secondary-60K35.

Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived Markovian coupled process (ξ t , ζ t ) t≥0 satisfies:(A) if ξ 0 ≤ ζ 0 (coordinate-wise), then for all t ≥ 0, ξ t ≤ ζ t a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on Z d such that, in each transition, k particles may jump from a site x to another site y, with k ≥ 1. These models include simple exclusion for which k = 1, but, beyond that value, the basic coupling construction is not possible and a more refined one is required. We give necessary and sufficient conditions on the rates to insure attractiveness; we construct a Markovian coupled process which both satisfies (A) and makes discrepancies between its two marginals non-increasing. We determine the extremal invariant and translation invariant probability measures under general irreducibility conditions. We apply our results to examples including a two-species asymmetric exclusion process with charge conservation (for which k ≤ 2) which arises from a Solid-on-Solid interface dynamics, and a stick process (for which k is unbounded) in correspondence with a generalized discrete Hammersley-Aldous-Diaconis model. We derive the hydrodynamic limit of these two one-dimensional models.

36 pagesInternational audienceWe 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

We establish necessary and sufficient conditions for weak convergence to the upper invariant measure for one-dimensional asymmetric nearest-neighbour zero-range processes with non-homogeneous jump rates. The class of "environments" considered is close to that considered by [1], while our class of processes is broader. We also give in arbitrary dimension a simpler proof of the result of [19] with weaker assumptions.