# Couplings, attractiveness and hydrodynamics for conservative particle systems

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

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

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topGobron, 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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