Couplings, attractiveness and hydrodynamics for conservative particle systems

Thierry Gobron; Ellen Saada

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 1132-1177
  • ISSN: 0246-0203

Abstract

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Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived markovian coupled process (ξt, ζt)t≥0 satisfies: (A) if ξ0≤ζ0 (coordinate-wise), then for all t≥0, ξt≤ζt a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y, with k≥1. These models include simple exclusion for which k=1, but, beyond that value, the basic coupling construction is not possible and a more refined one is required. We give necessary and sufficient conditions on the rates to insure attractiveness; we construct a markovian coupled process which both satisfies (A) and makes discrepancies between its two marginals non-increasing. We determine the extremal invariant and translation invariant probability measures under general irreducibility conditions. We apply our results to examples including a two-species asymmetric exclusion process with charge conservation (for which k≤2) which arises from a solid-on-solid interface dynamics, and a stick process (for which k is unbounded) in correspondence with a generalized discrete Hammersley–Aldous–Diaconis model. We derive the hydrodynamic limit of these two one-dimensional models.

How to cite

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Gobron, Thierry, and Saada, Ellen. "Couplings, attractiveness and hydrodynamics for conservative particle systems." Annales de l'I.H.P. Probabilités et statistiques 46.4 (2010): 1132-1177. <http://eudml.org/doc/241012>.

@article{Gobron2010,
abstract = {Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived markovian coupled process (ξt, ζt)t≥0 satisfies: (A) if ξ0≤ζ0 (coordinate-wise), then for all t≥0, ξt≤ζt a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y, with k≥1. These models include simple exclusion for which k=1, but, beyond that value, the basic coupling construction is not possible and a more refined one is required. We give necessary and sufficient conditions on the rates to insure attractiveness; we construct a markovian coupled process which both satisfies (A) and makes discrepancies between its two marginals non-increasing. We determine the extremal invariant and translation invariant probability measures under general irreducibility conditions. We apply our results to examples including a two-species asymmetric exclusion process with charge conservation (for which k≤2) which arises from a solid-on-solid interface dynamics, and a stick process (for which k is unbounded) in correspondence with a generalized discrete Hammersley–Aldous–Diaconis model. We derive the hydrodynamic limit of these two one-dimensional models.},
author = {Gobron, Thierry, Saada, Ellen},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {conservative particle systems; attractiveness; couplings; discrepancies; macroscopic stability; hydrodynamic limit; misanthrope process; discrete Hammersley–Aldous–Diaconis process; Stick process; solid-on-solid interface dynamics; two-species exclusion model; discrete Hammersley-Aldous-Diaconis process; stick process},
language = {eng},
number = {4},
pages = {1132-1177},
publisher = {Gauthier-Villars},
title = {Couplings, attractiveness and hydrodynamics for conservative particle systems},
url = {http://eudml.org/doc/241012},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Gobron, Thierry
AU - Saada, Ellen
TI - Couplings, attractiveness and hydrodynamics for conservative particle systems
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 1132
EP - 1177
AB - Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived markovian coupled process (ξt, ζt)t≥0 satisfies: (A) if ξ0≤ζ0 (coordinate-wise), then for all t≥0, ξt≤ζt a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y, with k≥1. These models include simple exclusion for which k=1, but, beyond that value, the basic coupling construction is not possible and a more refined one is required. We give necessary and sufficient conditions on the rates to insure attractiveness; we construct a markovian coupled process which both satisfies (A) and makes discrepancies between its two marginals non-increasing. We determine the extremal invariant and translation invariant probability measures under general irreducibility conditions. We apply our results to examples including a two-species asymmetric exclusion process with charge conservation (for which k≤2) which arises from a solid-on-solid interface dynamics, and a stick process (for which k is unbounded) in correspondence with a generalized discrete Hammersley–Aldous–Diaconis model. We derive the hydrodynamic limit of these two one-dimensional models.
LA - eng
KW - conservative particle systems; attractiveness; couplings; discrepancies; macroscopic stability; hydrodynamic limit; misanthrope process; discrete Hammersley–Aldous–Diaconis process; Stick process; solid-on-solid interface dynamics; two-species exclusion model; discrete Hammersley-Aldous-Diaconis process; stick process
UR - http://eudml.org/doc/241012
ER -

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