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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Recurrent random walk of an infinite particle system

Author: Frank Spitzer
Journal: Trans. Amer. Math. Soc. 198 (1974), 191-199
MSC: Primary 60K35
MathSciNet review: 0375533
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Abstract: Let $ p(x,y)$ be the transition function for a symmetric irreducible recurrent Markov chain on a countable set $ S$. Let $ {\eta _t}$ be the infinite particle system on $ S$ moving according to simple exclusion interaction with the one particle motion determined by $ p$. Assume that $ p$ is such that any two particles moving independently on $ S$ will sooner or later meet. Then it is shown that every invariant measure for $ {\eta _t}$ is a convex combination of Bernoulli product measures $ {\mu _\alpha }$ on $ {\{ 0,1\} ^s}$ with density $ 0 \leqslant \alpha = \mu [\eta (x) = 1] \leqslant 1$. Ergodic theorems are proved concerning the convergence of the system to one of the $ {\mu _\alpha }$.

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Keywords: Infinite particle systems, invariant measures, random walk, ergodic theorems
Article copyright: © Copyright 1974 American Mathematical Society