Invariant measures for multilane exclusion process

Abstract

We consider the simple exclusion process on Z × {0, 1}, that is, an "horizontal ladder" composed of 2 lanes. Particles can jump according to a lane-dependent translation-invariant nearest neighbour jump kernel, i.e. "horizontally" along each lane, and "vertically" along the scales of the ladder. We prove that generically, the set of extremal invariant measures consists of (i) translation-invariant product Bernoulli measures; and, modulo translations along Z: (ii) at most two shock measures (i.e. asymptotic to Bernoulli measures at ±∞) with asymptotic densities 0 and 2; (iii) at most three shock measures with a density jump of magnitude 1. We fully determine this set for certain parameter values. In fact, outside degenerate cases, there is at most one shock measure of type (iii). The result can be partially generalized to vertically cyclic ladders with arbitrarily many lanes. For the latter, we answer an open question of [5] about rotational invariance of stationary measures.

Ergodic theory of multi-layer interacting particle systems

“…We say that the dual process admits a successful coupling if for all n ∈ N, ξ, ξ ′ ∈ Ω n there exists a coupling {(ξ (1) (t), ξ (2) (t)) : t ≥ 0} of the processes {ξ(t) : t ≥ 0} starting from ξ and ξ ′ such that the following stopping time…”

“…where P ξ,ξ ′ is the path space measure of {(ξ (1) (t), ξ (2) (t)) : t ≥ 0} starting from ξ and ξ ′ .…”

“…Let Y (1) (t) and Y (2) (t) be two configurations of simple symmetric independent random walkers on the multi-layer state space N V . PROPOSITION 4.1.…”

“…PROOF. Since the particles move independently, by the Ornstein coupling we only have to show that there exists a successful coupling of two independent random walkers (Y (1) (t), σ (1) (t)) and (Y (2) (t), σ (2) (t)). First we define the stopping time ς = inf {t ≥ 0 : σ (1) (t) = σ (2) (t)}.…”

“…The characterization of the invariant measures of multi-layer exclusion processes has been obtained recently in [1]. The study of the hydrodynamic limit of a system of active particles has been studied in [8], and for a two-layer system with duality in [5].…”