On the Global Markov Property

  • Hans Föllmer


In a lattice model with nearest neighbor interaction, the Gibbs measures P have the local but not necessarily the global Markov property. We show that the global Markov property does hold in the following two cases: (1) The interaction satisfies Dobrushin’s unigueness condition, or (2) the interaction is attractive, and P is a “high density State”.


Gibbs Measure Markov Property Arbitrary Subset High Density State Standard Borel Space 
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  1. [1]
    Albeverio, S., Hoegh-Krohn, R., Olsen, G.: The global Markov property for lattice systems, Preprint Bielefeld, (Nov. 1978)Google Scholar
  2. [2]
    Dang-Ngoc, N Yor, M. Champs markoviens et mesures de Gibbs sur R. Ann. Scient. Ec. Nonn. Sup. 11, 29–69 (1978)MATHMathSciNetGoogle Scholar
  3. [3]
    Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions for its regularity. Th.Prob.Appl. 13, 197–224 (1968)CrossRefGoogle Scholar
  4. [4]
    Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Th.Prob.Appl. 15, 459–486 (1970)CrossRefGoogle Scholar
  5. [5]
    Föllmer, H.: Phase transition and Martin boundary. Sem. Prob. Strasbourg IX, Lecture Notes Math. 465, 305–317 (1975)Google Scholar
  6. [6]
    Föllmer, H.: On the Potential theory of stochastic fields. Proc. 40th session ISI, Warsaw (1975)Google Scholar
  7. [7]
    Preston, C.: Random fields. Lecture Notes Math. 534 (1976)MATHGoogle Scholar
  8. [8]
    Spitzer, F.: Phase transition in one dimensional nearest neighbor systems. J. Funct. Anal. 20, 240–254 (1975)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag/Wien 1980

Authors and Affiliations

  • Hans Föllmer
    • 1
  1. 1.MathematikdepartementETH-ZentrumZürichSwitzerland

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