Abstract
We investigate the connections of the Gibbs measures, which appear in Euclidean Field Theory, and the corresponding partial differential equations of Classical Euclidean Field Theory.
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Albeverio, S., Høegh-Krohn, R.: Uniqueness and the global Markov property for euclidean fields: the case of trigonometric interactions. Commun. Math. Phys.68, 95–127 (1979)
Dobrushin, R.L., Minlos, R.A.: Investigations of the properties of generalized Gaussian random fields. Sel. Math. Sov.1, 215–263 (1981)
Donald, M.: The classical field limit ofP(ϕ)2 quantum field theory. Commun. Math. Phys.79, 153–165 (1981)
Englisch, H.: Remarks on McBryan's convergence proof for (α−4(cos(αϕ)-1+α2ϕ2/2))-quantum fields.Rep. Math. Phys.18, 378–397 (1980)
Föllmer, H.: Phase transition and Martin boundary. Lecture Notes in Mathematics, Vol.465, pp. 305–317. Berlin, Heidelberg, New York: Springer
Glimm, J., Jaffe, A.: Quantum physics: A functional integral point of view. Berlin, Heidelberg, New York: Springer 1981
Goldstein, S.: Remarks on the global Markov property. Commun. Math. Phys.74, 223–234 (1980)
Holley, R., Strook, D.: The D.R.L. conditions for translation invariant Gaussian measures onL(R d). Z. Wahrscheinlichkeitstheor. Verw. Geb.53, 293–304 (1980)
Imbrie, J.Z.: Phase diagrams and cluster expansions for low temperatureP(ϕ)2 models. Commun. Math. Phys.82, 305–343 (1981)
McBryan, O.A.: The ϕ 42 quantum fields as a limit of sine-Gordon fields. Commun. Math. Phys.61, 275–284 (1978)
Nelson, E.: Probability theory and Euclidean field theory. In: Lecture Notes in Physics, Vol.25, pp. 94–124. Berlin, Heidelberg, New York: Springer 1973
Preston, C.J.: Random fields. Lecture Notes in Mathematics, Vol.534. Berlin, Heidelberg, New York: Springer 1975
Rauch, J., Williams, D.N.: Euclidean nonlinear classical field equations with unique vacuum. Commun. Math. Phys.63, 13–29 (1978)
Röckner, M.: A Dirichlet problem for distributions and the construction of specifications for Gaussian generalized random fields. Mem. Am. Math. Soc.54, 324 (1985)
Röckner, M.: Specifications and Martin Boundaries forP(φ)2-random fields. BiBoS No 126
Rozanov, Yu.A.: Markov random fields. Berlin, Heidelberg, New York: Springer 1982
Sinai, Ya.G.: Theory of phase transitions: rigorous results. Moscow: Nauka 1980
Smart, D.R.: Fixed point theorems. Cambridge, UK: Cambridge University Press 1974
Yosida, K.: Functional analysis. Berlin, Heidelberg, New York: Springer 1965
Zegarliński, B.: Uniqueness and the global Markov property for Euclidean fields: the case of general exponential interactions. Commun. Math. Phys.96, 195–221 (1984)
Zegarliński, B.: On the structure of Gibbs measure theory for Euclidean fields on lattice (in preparation)
Röckner, M., Zegarliński, B.: The Dirichlet problem for quasi-linear partial differential operators with boundary data given by distribution, to appear in Proc. of the 3rd BiBoS Symposium: Stochastic Processes Mathematics and Physics, Bielefeld Dec. 1985. Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer 1986
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Communicated by K. Osterwalder
On leave of absence from Institute of Theoretical Physics, University of Wroclaw, Wroclaw, Poland
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Zegarliński, B. The Gibbs measures and partial differential equations. Commun.Math. Phys. 107, 411–429 (1986). https://doi.org/10.1007/BF01220997
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DOI: https://doi.org/10.1007/BF01220997