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

Author:
Thomas M. Liggett

Journal:
Trans. Amer. Math. Soc. **179** (1973), 433-453

MSC:
Primary 60K35

DOI:
https://doi.org/10.1090/S0002-9947-1973-0326867-1

MathSciNet review:
0326867

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the transition function for a symmetric, irreducible, transient Markov chain on the countable set *S*. Let 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 in terms of the bounded functions on *S* which are harmonic with respect to . Ergodic theorems are proved concerning the convergence of the system to an invariant measure.

**[1]**Gustave Choquet and Jacques Deny,*Sur l’équation de convolution 𝜇=𝜇∗𝜎*, C. R. Acad. Sci. Paris**250**(1960), 799–801 (French). MR**0119041****[2]**William Feller,*An introduction to probability theory and its applications. Vol. II*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0210154****[3]**Richard Holley,*Free energy in a Markovian model of a lattice spin system*, Comm. Math. Phys.**23**(1971), 87–99. MR**0292449****[4]**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****[5]**Richard Holley,*An ergodic theorem for interacting systems with attractive interactions*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**24**(1972), 325–334. MR**0331562**, https://doi.org/10.1007/BF00679137**[6]**Thomas M. Liggett,*Existence theorems for infinite particle systems*, Trans. Amer. Math. Soc.**165**(1972), 471–481. MR**0309218**, https://doi.org/10.1090/S0002-9947-1972-0309218-7**[7]**Thomas M. Liggett,*An infinite particle system with zero range interactions*, Ann. Probability**1**(1973), 240–253. MR**0381039**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
60K35

Retrieve articles in all journals with MSC: 60K35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0326867-1

Keywords:
Infinite particle systems,
invariant measures,
ergodic theorems

Article copyright:
© Copyright 1973
American Mathematical Society