Stationary blocking measures for one-dimensional nonzero mean exclusion processes

Convergence and local equilibrium for the one-dimensional nonzero mean exclusion process

“…In Section 2, we introduce the main tools that will be used in the proof of Theorem 1: (i) coupling and priority rules; (ii) the finite propagation property; (iii) the Burgers-type hydrodynamic limit for finiterange nonzero mean exclusion processes, proved by Rezakhanlou ([13]), and its extension to arbitrary initial conditions, which follows from [2]. In Section 3 we prove the following preliminary statement: at times of order N → ∞, the number of discrepancies in an interval of order N between η .…”

“…Theorem 7 is a straightforward consequence of Proposition 3.1 from [2]. For completeness, the proof is detailed in Appendix A.…”

“…The key argument is the following lemma, whose proof is contained (up to a slight difference in the definition of T i j ) in the proof of Lemma 3.2 in [2]. It relies on Lemma 5 and strong Markov property.…”

Constructive Euler Hydrodynamics for One-Dimensional Attractive Particle Systems

“…The irreducibility assumption (that is (8) in the case of the misanthrope's process) plays an essential role there. To take advantage of [17], we treat separately the movements of discrepancies corresponding to big jumps, which can be controlled thanks to the finiteness of the first moment (that is, assumption (M4) on page 10).…”

Equilibrium interfaces of biased voter models

“…Other models for which this has been proved include one-dimensional two-type contact processes with strong bias (in the sense that one type can overtake the other, but not vice versa) [AMPV10], or no bias (no type can infect a site occupied by the other type) [Val10,MV16], as well as one-dimensional asymmetric exclusion processes that admit so-called blocking measures (see e.g. [BM02,BLM02]).…”