The Brownian web (BW) is the random network formally consisting of the paths of coalescing one-dimensional Brownian motions starting from every space-time point in R\timesR. We extend the earlier work of Arratia and of Toth and Werner by providing a new characterization which is then used to obtain convergence results for the BW distribution, including convergence of the system of all coalescing random walks to the BW under diffusive space-time scaling.Comment: Published at http://dx.doi.org/10.1214/009117904000000568 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time R×R. It was first introduced by Arratia, and later analyzed in detail by Tóth and Werner. More recently, Fontes, Isopi, Newman and Ravishankar (FINR) gave a characterization of the BW, and general convergence criteria allowing in principle either crossing or noncrossing paths, which they verified for coalescing simple random walks. Later Ferrari, Fontes, and Wu verified these criteria for a two dimensional Poisson Tree. In both cases, the paths are noncrossing. To date, the general convergence criteria of FINR have not been verified for any case with crossing paths, which appears to be significantly more difficult than the noncrossing paths case. Accordingly, in this paper, we formulate new convergence criteria for the crossing paths case, and verify them for non-simple coalescing random walks satisfying a finite fifth moment condition. This is the first time that convergence to the BW has been proved for models with crossing paths. Several corollaries are presented, including an analysis of the scaling limit of voter model interfaces that extends a result of Cox and Durrett.

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.

The dynamical discrete web (DyDW), introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter τ . The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed τ . In this paper, we study the existence of exceptional (random) values of τ where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of exceptional such τ . Our results are motivated by those about exceptional times for dynamical percolation in high dimension by Häggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in the DyDW is rather different from the situation for the dynamical random walks of Benjamini, Häggstrom, Peres and Steif.For example, we prove that the walk from the origin S τ 0 violates the law of the iterated logarithm (LIL) on a set of τ of Hausdorff dimension one. We also discuss how these and other results extend to the dynamical Brownian web, the natural scaling limit of the DyDW.