Ergodic and mixing properties of equilibrium measures for Markov processes
Author:
Enrique D. Andjel
Journal:
Trans. Amer. Math. Soc. 318 (1990), 601614
MSC:
Primary 60J25; Secondary 28D05, 60K35
MathSciNet review:
953535
Fulltext PDF Free Access
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]
Michael
Aizenman, Translation invariance and instability of phase
coexistence in the twodimensional Ising system, Comm. Math. Phys.
73 (1980), no. 1, 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, 197224.
 [3]
, (1970), Prescribing a system of random variables by the help of conditional distributions, Theory Probab. Appl. 15, 458486.
 [4]
R.
L. Dobrushin and S.
B. Shlosman, Constructive criterion for the uniqueness of Gibbs
field, Statistical physics and dynamical systems (Köszeg, 1984)
Progr. Phys., vol. 10, Birkhäuser Boston, Boston, MA, 1985,
pp. 347–370. MR 821306
(87d:82006)
 [5]
E.
B. Dynkin, Sufficient statistics and extreme points, Ann.
Probab. 6 (1978), no. 5, 705–730. MR 0518321
(58 #24575)
 [6]
Stewart
N. Ethier and Thomas
G. Kurtz, Markov processes, Wiley Series in Probability and
Mathematical Statistics: Probability and Mathematical Statistics, John
Wiley & Sons, Inc., New York, 1986. Characterization and convergence.
MR 838085
(88a:60130)
 [7]
H.
Föllmer, A covariance estimate for Gibbs measures, J.
Funct. Anal. 46 (1982), no. 3, 387–395. MR 661878
(84d:60142), http://dx.doi.org/10.1016/00221236(82)900532
 [8]
Y.
Higuchi, On the absence of nontranslation invariant Gibbs states
for the twodimensional Ising model, Random fields, Vol. I, II
(Esztergom, 1979) Colloq. Math. Soc. János Bolyai, vol. 27,
NorthHolland, AmsterdamNew York, 1981, pp. 517–534. MR 712693
(84m:82020)
 [9]
Yasunari
Higuchi and Tokuzo
Shiga, Some results on Markov processes of infinite lattice spin
systems, J. Math. Kyoto Univ. 15 (1975),
211–229. MR 0370831
(51 #7056)
 [10]
Richard
Holley, Free energy in a Markovian model of a lattice spin
system, Comm. Math. Phys. 23 (1971), 87–99. MR 0292449
(45 #1535)
 [11]
R.
Holley and D.
Stroock, Diffusions on an infinitedimensional torus, J.
Funct. Anal. 42 (1981), no. 1, 29–63. MR 620579
(82k:60152), http://dx.doi.org/10.1016/00221236(81)900471
 [12]
Thomas
M. Liggett, Interacting particle systems, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 276, SpringerVerlag, New York, 1985. MR 776231
(86e:60089)
 [13]
Norman
S. Matloff, Ergodicity conditions for a dissonant voting
model, Ann. Probability 5 (1977), no. 3,
371–386. MR 0445646
(56 #3982)
 [14]
David
Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics
and its Applications, vol. 5, AddisonWesley Publishing Co., Reading,
Mass., 1978. The mathematical structures of classical equilibrium
statistical mechanics; With a foreword by Giovanni Gallavotti and
GianCarlo Rota. MR 511655
(80g:82017)
 [1]
 M. Aizenman (1980), Translation invariance and instability of phase coexistence in the two dimensional Ising system, Comm. Math. Phys. 73, 8394. 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, 197224.
 [3]
 , (1970), Prescribing a system of random variables by the help of conditional distributions, Theory Probab. Appl. 15, 458486.
 [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. 347370. MR 821306 (87d:82006)
 [5]
 E. B. Dynkin (1978), Sufficient statistics and extreme points, Ann. Probab. 6, 705730. 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, 387395. MR 661878 (84d:60142)
 [8]
 Y. Higuchi (1979), On the absence of nontranslation invariant Gibbs states for the two dimensional Ising model, Colloq. Math. Soc. János Bolyai 27, 517533. 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, 211229. MR 0370831 (51:7056)
 [10]
 R. Holley (1971), Free energy in a Markovian model of a lattice spin system, Comm. Math. Phys. 23, 8799. MR 0292449 (45:1535)
 [11]
 R. Holley and D. Stroock (1981), Diffusions on an infinitedimensional torus, J. Funct. Anal. 42, 2963. MR 620579 (82k:60152)
 [12]
 T. M. Liggett (1985), Interacting particle systems, SpringerVerlag, New York. MR 776231 (86e:60089)
 [13]
 N. Matloff (1977), Ergodicity conditions for a dissonant voting model, Ann. Probab. 5, 371386. MR 0445646 (56:3982)
 [14]
 D. Ruelle (1978), Thermodynamic formalism, AddisonWesley, Reading, Mass. MR 511655 (80g:82017)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
60J25,
28D05,
60K35
Retrieve articles in all journals
with MSC:
60J25,
28D05,
60K35
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009535355
PII:
S 00029947(1990)09535355
Keywords:
Markov process,
interacting particle system,
ergodic,
mixing
Article copyright:
© Copyright 1990
American Mathematical Society
