Consider a continuous time particle system η$$ = (η$$(k), k ∈ 𝕃), indexed by a lattice 𝕃 which will be either ℤ, ℤ∕nℤ, a segment {1, ⋯ , n}, or ℤ$$, and taking its values in the set $$ where $$ = {0, ⋯ , κ − 1} for some fixed κ ∈{∞, 2, 3, ⋯ }. Assume that the Markovian evolution of the particle system (PS) is driven by some translation invariant local dynamics with bounded range, encoded by a jump rate matrix ⊤. These are standard settings, satisfied by the TASEP, the voter models, the contact processes. The aim of this paper is to provide some sufficient and/or necessary conditions on the matrix ⊤ so that this Markov process admits some simple invariant distribution, as a product measure (if 𝕃 is any of the spaces mentioned above), the law of a Markov process indexed by ℤ or [1, n] ∩ ℤ (if 𝕃 = ℤ or {1, …, n}), or a Gibbs measure if 𝕃 = ℤ/nℤ. Multiple applications follow: efficient ways to find invariant Markov laws for a given jump rate matrix or to prove that none exists. The voter models and the contact processes are shown not to possess any Markov laws as invariant distribution (for any memory m). (As usual, a random process X indexed by ℤ or ℕ is said to be a Markov chain with memory m ∈ {0, 1, 2, ⋯ } if ℙ(X$$ ∈ A | X$$, i ≥ 1) = ℙ(X$$ ∈ A | X$$, 1 ≤ i ≤ m), for any k.) We also prove that some models close to these models do. We exhibit PS admitting hidden Markov chains as invariant distribution and design many PS on ℤ2, with jump rates indexed by 2 × 2 squares, admitting product invariant measures.
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/ps/2020008
Mots-clés : Particle systems, invariant distributions
@article{PS_2020__24_1_526_0,
author = {Fredes, Luis and Marckert, Jean-Fran\c{c}ois},
title = {Invariant measures of interacting particle systems: {Algebraic} aspects},
journal = {ESAIM: Probability and Statistics},
pages = {526--580},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2020008},
mrnumber = {4160331},
zbl = {1455.60102},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2020008/}
}
TY - JOUR AU - Fredes, Luis AU - Marckert, Jean-François TI - Invariant measures of interacting particle systems: Algebraic aspects JO - ESAIM: Probability and Statistics PY - 2020 SP - 526 EP - 580 VL - 24 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2020008/ DO - 10.1051/ps/2020008 LA - en ID - PS_2020__24_1_526_0 ER -
%0 Journal Article %A Fredes, Luis %A Marckert, Jean-François %T Invariant measures of interacting particle systems: Algebraic aspects %J ESAIM: Probability and Statistics %D 2020 %P 526-580 %V 24 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2020008/ %R 10.1051/ps/2020008 %G en %F PS_2020__24_1_526_0
Fredes, Luis; Marckert, Jean-François. Invariant measures of interacting particle systems: Algebraic aspects. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 526-580. doi : 10.1051/ps/2020008. http://archive.numdam.org/articles/10.1051/ps/2020008/
[1] and , An introduction to Gröbner bases, in Vol. 3 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1994). | MR | Zbl
[2] , Invariant measures for the zero range processes. Ann. Probab. 10 (1982) 525–547. | DOI | MR | Zbl
[3] , The stationary measure of a 2-type totally asymmetric exclusion process. J. Combin. Theory Ser. A 113 (2006) 4. | DOI | MR | Zbl
[4] , , and , Existence of the zero range process and a deposition model with superlinear growth rates. Ann. Probab. 35 (2007) 1201–1249. | DOI | MR | Zbl
[5] and , Nonequilibrium steady states of matrix-product form: a solver’s guide. J. Phys. A 40 (2007) R333–R441. | DOI | MR | Zbl
[6] , and , Inference in hidden Markov models. Springer Science & Business Media, Berlin (2006). | MR | Zbl
[7] and , Markovianity of the invariant distribution of probabilistic cellular automata on the line. Stoch. Process Appl. 125 (2015) 3458–3483. | DOI | MR | Zbl
[8] , and , Integrable approach to simple exclusion processes with boundaries. Review and progress. J. Stat. Mech. Theory Exp. 11 (2014) P11032. | DOI | MR | Zbl
[9] , and , Stationary Measures and Phase Transition for a Class of Probabilistic Cellular Automata. ESAIM: PS 6 (2002) 89–104. | DOI | Numdam | MR | Zbl
[10] , , and , Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26 (1993) 1493–1517. | DOI | MR | Zbl
[11] , , , , , , and . Phase transition in 2d partially asymmetric simple exclusion process with two species. J. Stat. Mech. Theory Exp. 2014 (2014) P10002. | DOI | MR | Zbl
[12] , and , Factorized steady states in mass transport models. J. Phys. A 37 (2004) L275–L280. | DOI | MR | Zbl
[13] , and , Invariant measures of mass migration processes. Electron. J. Probab. 21 (2016) 60. | DOI | MR | Zbl
[14] , Personal page. Available from: https://www-polsys.lip6.fr/~jcf/ (2020).
[15] and , Maple file and pdf file. Available at: http://www.labri.fr/perso/marckert/Grobner.mw, http://www.labri.fr/perso/marckert/Grobner.pdf (2020).
[16] , Gibbs Measures and Phase Transitions, Series:De Gruyter Studies in Mathematics 9, De Gruyter, Berlin (2011). | DOI | MR | Zbl
[17] and , Product measure steady states of generalized zero range processes. J. Phys. A 39 (2006) 1565–1573. | DOI | MR | Zbl
[18] , Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9 (1972) 66–89. | DOI | MR | Zbl
[19] and , Scaling limits of interacting particle systems, Vol. 320 of Fundamental Principles of Mathematical Sciences. Springer-Verlag, Berlin (1999). | MR | Zbl
[20] , Stationary product measures for conservative particle systems and ergodicity criteria. Electron. J. Probab. 18 (2013) 88. | DOI | MR | Zbl
[21] , Interacting particle systems, Classics in Mathematics. Springer-Verlag, Berlin (2005). | MR | Zbl
[22] and , Probabilistic cellular automata and random fields with iid directions. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014) 455–475. | DOI | Numdam | MR | Zbl
[23] , A Course in Interacting Particle Systems (2017).
[24] , , , , and , Stochastic cellular systems: ergodicity, memory, morphogenesis (Part: Discrete local Markov systems. Manchester University Press, Manchester (1990), 1–182.
[25] , Bernoulli and Markov stationary measures in discrete local interactions, Vol. 1 of Developments in Statistics. Academic Press, New York (1978). | MR | Zbl
[26] and , Reversible Markov chains with local interactions. Adv. Probab. Related Topics 6 (1980) 451–469. | MR | Zbl
Cité par Sources :
