Ergodic and mixing properties of equilibrium measures for Markov processes

Author:
Enrique D. Andjel

Journal:
Trans. Amer. Math. Soc. **318** (1990), 601-614

MSC:
Primary 60J25; Secondary 28D05, 60K35

DOI:
https://doi.org/10.1090/S0002-9947-1990-0953535-5

MathSciNet review:
953535

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the semigroup corresponding to a Markov process on a metric space . Suppose commutes with a homeomorphism of . We prove that under certain conditions, an equilibrium measure for the process is ergodic under . We also show that, under stronger conditions this measure must be mixing under . Several applications of these results are given.

**[1]**M. Aizenman (1980),*Translation invariance and instability of phase coexistence in the two dimensional Ising system*, Comm. Math. Phys.**73**, 83-94. MR**573615 (82c:82003)****[2]**R. L. Dobrushin (1968),*The description of a random field by means of conditional probabilities and conditions of regularity*, Theory Probab. Appl.**13**, 197-224.**[3]**-, (1970),*Prescribing a system of random variables by the help of conditional distributions*, Theory Probab. Appl.**15**, 458-486.**[4]**R. L. Dobrushin and S. B. Shlosman (1985),*Constructive criterion for the uniqueness of Gibbs field*, Statistical Physics and Dynamical Systems (J. Fritz, A. Jaffe and D. Szasz, eds), Birkhäuser, Boston, Mass., pp. 347-370. MR**821306 (87d:82006)****[5]**E. B. Dynkin (1978),*Sufficient statistics and extreme points*, Ann. Probab.**6**, 705-730. MR**0518321 (58:24575)****[6]**S. N. Ethier and T. G. Kurtz (1986),*Markov processes: characterization and convergence*, Wiley, New York. MR**838085 (88a:60130)****[7]**H. Föllmer (1982),*A covariance estimate for Gibbs measures*, J. Funct. Anal.**46**, 387-395. MR**661878 (84d:60142)****[8]**Y. Higuchi (1979),*On the absence of non-translation invariant Gibbs states for the two dimensional Ising model*, Colloq. Math. Soc. János Bolyai**27**, 517-533. MR**712693 (84m:82020)****[9]**Y. Higuchi and T. Shiga (1975),*Some results on Markov processes of infinite lattice spin systems*, J. Math. Kyoto Univ.**15**, 211-229. MR**0370831 (51:7056)****[10]**R. Holley (1971),*Free energy in a Markovian model of a lattice spin system*, Comm. Math. Phys.**23**, 87-99. MR**0292449 (45:1535)****[11]**R. Holley and D. Stroock (1981),*Diffusions on an infinite-dimensional torus*, J. Funct. Anal.**42**, 29-63. MR**620579 (82k:60152)****[12]**T. M. Liggett (1985),*Interacting particle systems*, Springer-Verlag, New York. MR**776231 (86e:60089)****[13]**N. Matloff (1977),*Ergodicity conditions for a dissonant voting model*, Ann. Probab.**5**, 371-386. MR**0445646 (56:3982)****[14]**D. Ruelle (1978),*Thermodynamic formalism*, Addison-Wesley, Reading, Mass. MR**511655 (80g:82017)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-0953535-5

Keywords:
Markov process,
interacting particle system,
ergodic,
mixing

Article copyright:
© Copyright 1990
American Mathematical Society