Ergodic theorem for generalized long-range exclusion processes with positive recurrent transition probabilities.

*(English)*Zbl 0657.60126Let S be a set of sites, finite or countable. Consider a configuration of particles in these sites, with at most m particles at each site, m bigger than a fixed number. The k particles at site \(s\in S\) wait for an exponential period of time of parameter g(k), independent of all other sites and then a particle begins to move according to a Markov chain rule U on the sites until it finds a site which has less than m particles. This search is instantaneous and is not time consuming.

The author proved that there is a Markov process defined on the set of configurations which is a model for the above heuristic description. In the case that the associated Markov chain U is positive recurrent, then for each \(0\leq n<\infty\) there is an ergodic stationary measure for configuration with a total of n particles. And for an infinite number of particles, the only invariant measure is with all sites occupied with m particles. Also any probability measure on the configurations will converge weakly to a mixture of the above mentioned invariant measures.

The author proved that there is a Markov process defined on the set of configurations which is a model for the above heuristic description. In the case that the associated Markov chain U is positive recurrent, then for each \(0\leq n<\infty\) there is an ergodic stationary measure for configuration with a total of n particles. And for an infinite number of particles, the only invariant measure is with all sites occupied with m particles. Also any probability measure on the configurations will converge weakly to a mixture of the above mentioned invariant measures.

Reviewer: M.Smorodinsky

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60F05 | Central limit and other weak theorems |

60J25 | Continuous-time Markov processes on general state spaces |

##### Keywords:

exclusion processes; infinite particle system; ergodic stationary measure; invariant measure
PDF
BibTeX
XML
Cite

\textit{X. Zheng}, Acta Math. Sin., New Ser. 4, No. 3, 193--209 (1988; Zbl 0657.60126)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Chung, K.L., Markov chains with stationary transition probabilities, Springer Verlag, 1960. · Zbl 0092.34304 |

[2] | Holley, R., An ergodic theorem for interacting systems with attractive interactions,ZWVG.,14(1973), 325–334. · Zbl 0251.60066 |

[3] | Liggett, T.M., Convergence to total occupancy in an infinite particle system with interactions,Ann. prob.,2 (1974), 989–998. · Zbl 0295.60086 |

[4] | Liggett, T.M., Long range exclusion processes,Ann. prob.,8(1980), 861–889. · Zbl 0457.60079 |

[5] | Qian, M.P., The decomposition of a stationary Markov chain into a detailed balanced part and a circulation part,Scientia Sinica, 1979. · Zbl 0421.60061 |

[6] | Spitzer, F., Interaction of Markov processes,Advance in Math.,5(1970), 246–290. · Zbl 0312.60060 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.