Macroscopic stability for nonfinite range kernels

Abstract

We extend the strong macroscopic stability introduced in Bramson & Mountford (2002) for one-dimensional asymmetric exclusion processes with finite range to a large class of one-dimensional conservative attractive models (including misanthrope process) for which we relax the requirement of finite range kernels. A key motivation is extension of constructive hydrodynamics result of Bahadoran et al. (2002, 2006, 2008) to nonfinite range kernels

Euler hydrodynamics for attractive particle systems in random environment

“…Using Lemma 3.1,(iii), Lemmas 3.2 and 3.4, and Proposition 3.2, we obtain ν 1 ≪ ν 2 , that is (23). Existence (22) of an asymptotic particle density can be obtained by a proof analogous to [24,Lemma 14], where the space-time ergodic theorem is applied to the joint disorder-particle process. Then, closedness of R Q is established as in [4, Proposition 3.1].…”

Constructive Euler Hydrodynamics for One-Dimensional Attractive Particle Systems

“…Then, closedness of R Q is established as in [7, Proposition 3.1]: it uses (38), (37). Given the rest of the proposition, the weak continuity statement comes from a coupling argument, using (38) and Lemma 3.1.…”

“…In this paper, we review successive stages ( [6,7,8,38,9]) of a constructive approach to hydrodynamic limits given by equations of the type (1), which ultimately led us in [9] to a very general hydrodynamic limit result for attractive particle systems in one dimension in ergodic random environment.…”

“…The latter result, apart from its own interest, proved to be an essential step to obtain quenched hydrodynamics in the disordered case ( [9]). For this last paper, we also relied on [38], which improves an essential property we use, macroscopic stability.…”

“…In Section 2, after giving general notation and definitions, we introduce the two basic models we originally worked with, the misanthropes process and the k-step exclusion process. Then we describe informally the results, and the main ideas involved in [4] which was our starting point, and in each of the papers [6,7,8,38,9]. Section 3 contains our main results, stated (for convenience) for the misanthropes process.…”