A characterization of the invariant measures for an infinite particle system with interactions

Liggett, Thomas M.

doi:10.1090/S0002-9947-1973-0326867-1

A characterization of the invariant measures for an infinite particle system with interactions
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Trans. Amer. Math. Soc. 179 (1973), 433-453 Request permission

Abstract:

Let $p(x,y)$ be the transition function for a symmetric, irreducible, transient Markov chain on the countable set S. Let ${\eta _t}$ be the infinite particle system on S with the simple exclusion interaction and one-particle motion determined by p. A characterization is obtained of all the invariant measures for ${\eta _t}$ in terms of the bounded functions on S which are harmonic with respect to $p(x,y)$. Ergodic theorems are proved concerning the convergence of the system to an invariant measure.

References

Gustave Choquet and Jacques Deny, Sur l’équation de convolution $\mu =\mu \ast \sigma$, C. R. Acad. Sci. Paris 250 (1960), 799–801 (French). MR 119041
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Richard Holley, Free energy in a Markovian model of a lattice spin system, Comm. Math. Phys. 23 (1971), 87–99. MR 292449, DOI 10.1007/BF01877751
Richard Holley, Pressure and Helmholtz free energy in a dynamic model of a lattice gas, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 565–578. MR 0413308
Richard Holley, An ergodic theorem for interacting systems with attractive interactions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), 325–334. MR 331562, DOI 10.1007/BF00679137
Thomas M. Liggett, Existence theorems for infinite particle systems, Trans. Amer. Math. Soc. 165 (1972), 471–481. MR 309218, DOI 10.1090/S0002-9947-1972-0309218-7
Thomas M. Liggett, An infinite particle system with zero range interactions, Ann. Probability 1 (1973), 240–253. MR 381039, DOI 10.1214/aop/1176996977