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 a one-dimensional totally asymmetric nearest-neighbor zero-range process with site-dependent jump-rates -an environment. For each environment p we prove that the set of all invariant measures is the convex hull of a set of product measures with geometric marginals. As a consequence we show that for environments p satisfying certain asymptotic property, there are no invariant measures concentrating on configurations with density bigger than ρ * (p), a critical value. If ρ * (p) is finite we say that there is phase-transition on the density. In this case we prove that if the initial configuration has asymptotic density strictly above ρ * (p), then the process converges to the maximal invariant measure.

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

Published at http://dx.doi.org/10.1214/105051605000000151 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)International audienceWe prove a strong law of large numbers for the location of the second class particle in a totally asymmetric exclusion process when the process is started initially from a decreasing shock. This completes a study initiated in Ferrari and Kipnis [Ann. Inst. H. Poincare Probab. Statist. 13 (1995) 143-154]