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Uniqueness and global Markov property for Euclidean fields: The case of general polynomial interactions

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Abstract

We give a general method for proving uniqueness and global Markov property for Euclidean quantum fields. The method is based on uniform continuity of local specifications (proved by using potential theoretical tools) and exploitation of a suitable FKG-order structure. We apply this method to give a proof of uniqueness and global Markov property for the Gibbs states and to study extremality of Gibbs states also in the case of non-uniqueness. In particular we prove extremality for ϕ 42 (also in the case of non-uniqueness), and global Markov property for weak coupling ϕ 42 (which solves a long-standing problem). Uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field.

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References

  1. Accardi, L.: Local perturbations of conditional expectations. J. Math. Anal. Appl.72, 34–69 (1979)

    Google Scholar 

  2. Adler, R.J., Epstein, R.: Some central limit theorems for Markov paths and some properties of Gaussian random fields. Stoch. Processes Appl.24, 157–202 (1987)

    Google Scholar 

  3. Aizenman, M.: Translation invariance and instability of phase coexistence in the two dimensional Ising systems. Commun. Math. Phys.73, 83–94 (1980)

    Google Scholar 

  4. Albeverio, S., Høegh-Krohn, R.: Some recent interactions between mathematics and physics in connection with generalized random fields. In: Proc. 1st World Congress of the Bernoulli Society, Tashkent 1986, VNU-Press (1988)

  5. Albeverio, S., Blanchard, Ph., Høegh-Krohn, R.: Some applications of functional integration. In: Mathematical problems in theoretical physics. Proc. IAMP Conf. Berlin, 1981. Schrader, R., Seiler, R., Uhlenbeck, D.A. (eds.). Lecture Notes in Physisc. Vol.153, Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  6. Albeverio, S., Fenstad, J.E., Høegh-Krohn, R., Lindstrøm, T.: Nonstandard methods in stochastic analysis and mathematical physics. New York: Academic Press 1986

    Google Scholar 

  7. Albeverio, S., Høegh-Krohn, R.: The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time, J. Funct. Anal.16, 39–82 (1974)

    Google Scholar 

  8. Albeverio, S., Høegh-Krohn, R.: Quasi-invariant measures, symmetric diffusion processes and quantum fields, pp. 11–59. In: Proceedings of the International Colloquium on Mathematical Methods of Quantum Fields Theory. Guerra, F., Robinson, D.W., Stora, R. (eds.) CNRS No 248 (1976)

  9. Albeverio, S., Høegh-Krohn, R.: Dirichlet forms and diffusion processes on rigged Hilbert spaces. Z. Wahrscheinlichkeitstheorie Verw. Geb.40, 1–57 (1977)

    Google Scholar 

  10. Albeverio, S., Høegh-Krohn, R.: Hunt processes and analytic potentials theory on rigged Hilbert spaces. Ann. Inst. H. Poincaré B13, 269–291 (1977)

    Google Scholar 

  11. Albeverio, S., Høegh-Krohn, R.: Topics in infinite dimensional analyis, pp. 279–302. In: Mathematical problems in theoretical physics. Dell'Antonio, G., Doplicher, S., Jona-Lasinio, G. (eds.). Lectures Notes Physics. Vol.80. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  12. Albeverio, S., Høegh-Krohn, R.: The global Markov property for Euclidean and lattice fields. Phys. Lett.84 B, 89–90 (1979)

    Google Scholar 

  13. Albeverio, S., Høegh-Krohn, R.: Uniqueness and the global Markov property for Euclidean fields. The case of trigonometric interactions. Commun. Math. Phys.68, 95–128 (1979)

    Google Scholar 

  14. Albeverio, S., Høegh-Krohn, R.: Uniqueness of Gibbs states and global Markov property for Euclidean fields. Proc. Colloquium “Random fields — rigorous results in statistical mechanics and quantum fields theory.” Esztergom 1979, Fritz, J., Lebowitz, J.L., Szász, D. (eds.). Colloq. Math. Soc. J. Bolyai Vol.27. Amsterdam: North-Holland 1979

    Google Scholar 

  15. Albeverio, S., Høegh-Krohn, R.: Uniqueness and global Markov property for Euclidean fields and Lattice systems, pp. 303–320. In: Quantum fields-algebras-processes. Streit, L. (ed.). Wien: Springer 1980

    Google Scholar 

  16. Albeverio, S., Høegh-Krohn, R.: Markov processes and Markov fields in quantum theory, group theory, hydrodynamics and C*-algebras, pp. 497–540. In: Stochastic integrals. Williams, D. (ed.). Lecture Notes Mathematics Vol.851, Berlin, Heidelberg, New York: Springer 1981

    Google Scholar 

  17. Albeverio, S., Høegh-Krohn, R.: Stochastic methods in quantum field theory and hydrodynamics, pp. 193–215. In: New stochastic methods in physics. De Witt-Morette, C., Elworthy, K.D. (eds.). Phys. Rep.77, (1981)

  18. Albeverio, S., Høegh-Krohn, R.: The method of Dirichlet forms. In: Stochastic behavior in classical and quantum Hamiltonian systems. Casati, G., Ford, J. (eds.). Lecture Notes in Physics, vol. 93, pp. 250–258. Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  19. Albeverio, S., Høegh-Krohn, R.: Diffusion, quantum fields and groups of mappings, pp. 133–145. In: Functional analysis in Markov processes, Fukushima, M. (ed.). Lecture Notes in Maths. Vol.923, Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  20. Albeverio, S., Høegh-Krohn, R.: Local and global Markoff fields. Rep. Math. Phys.19, 225–248 (1984)

    Google Scholar 

  21. Albeverio, S., Høegh-Krohn, R.: Diffusion fields, quantum fields and fields with values in groups. In: Adv. in probability, stochastic analysis and applications. Pinsky, M. (ed.). New York: Dekker 1984

    Google Scholar 

  22. Albeverio, S., Høegh-Krohn, R., Holden, H.: Markov processes on infinite dimensional spaces, Markov fields and Markov cosurfaces, pp. 11–40. In: Stochastic space-time models and limit theorems. Arnold, L., Kotelenez, P. (eds.). Dordrecht: Reidel 1985

    Google Scholar 

  23. Albeverio, S., Høegh-Krohn, R., Olsen, G.: The global Markov property for Lattice fields. J. Multiv. Anal.11, 599–607 (1981)

    Google Scholar 

  24. Albeverio, S., Høegh-Krohn, R., Röckner, M.: Canonical relativistic local fields and Dirichlet forms (in preparation)

  25. Albeverio, S., Høegh-Krohn, R., Zegarlinski, B.: Uniqueness of Gibbs states for generalP(φ)2-weak coupling models by cluster expansion. Commun. Math. Phys.121, 683–697 (1989)

    Google Scholar 

  26. Albeverio, S., Kusuoka, S.: Maximality of Dirichlet forms on infinite dimensional spaces and Høegh-Krohn's model of quantum fields, SFB-Preprint(1988) to appear in book dedicated to the Memory of R. Høegh-Krohn

  27. Albeverio, S., Röckner, M.: Classical Dirichlet forms on topological vector spaces — Closability and a Cameron Martin formula, SFB 237-Preprint 13 (1988), to appear in J. Funct. Anal. (1989)

  28. Albeverio, S., Röckner, M.: Dirichlet forms, quantum fields and stochastic quantisation. In: Proc. Warwick Conf. Stochastic analysis, path integration and dynamics. Elworthy, K.D., Zambrini, J.-C. (eds.), pp. 1–21. London: Pitman (1989)

    Google Scholar 

  29. Araki, H.: Hamiltonian formalism and the canonical commutation relations in quantum field theory. J. Math. Phys.1, 492–504 (1960)

    Google Scholar 

  30. Atkinson, B.: On Dynkin's Markov property of random fields associated with symmetric processes. Stoch. Proc. Appl.15, 193–201 (1983)

    Google Scholar 

  31. Bellissard, J., Høegh-Krohn, R.: Compactness and the maximal Gibbs state for random Gibbs fields on a lattice. Commun. Math. Phys.84, 297–327 (1982)

    Google Scholar 

  32. Bellissard, J., Picco, P.: Lattice quantum fields: uniqueness and Markov property. Preprint Marseille (1979)

  33. Bers, L., John, F., Schechter, M.: Partial differential equations. New York: Interscience (1964)

    Google Scholar 

  34. Blanchard, Ph., Pfister, Ch.: Processus Gaussiens, équivalence d'ensembler et spécification locale. Z. Wahrscheinlichkeitstheor. Verw. Geb.44, 177–190 (1978)

    Google Scholar 

  35. Bricmont, J., Fontaine, J.R., Landau, J.L.: Absence of symmetry breakdown and uniqueness of the vacuum for multicomponent field theories. Commun. Math. Phys.64, 49–72 (1979)

    Google Scholar 

  36. Brydges, D., Fröhlich, J., Spencer, T.: The random walk representation of classical spin systems and correlation inequalities. Commun. Math. Phys.83, 123–150 (1982)

    Google Scholar 

  37. Cassandro, M., Olivieri, E., Pellegrinotti, E., Presutti, E.: Existence and uniqueness of DLR measures for unbounded spin systems. Z. Wahrscheinlichkeitstheor. Verw. Geb.41, 313–334 (1978)

    Google Scholar 

  38. Challifour, J.: Euclidean field theory. II Remarks on embedding the relativistic Hilbert space. J. Math. Phys.17, 1889–1892 (1976)

    Google Scholar 

  39. Chen, E.: Markoff systems of imprimitivity. J. Math. Phys.18, 1832–1837 (1977)

    Google Scholar 

  40. Coester, F., Haag, R.: Representation of states in a field theory with canonical variables. Phys. Rev.117, 1137–1145 (1960)

    Google Scholar 

  41. Dellacherie, C., Meyer, P.A.: Probabilités et potentiel, Ch. XII-XVI. Paris: Hermann 1987

    Google Scholar 

  42. Dimock, J., Glimm, J.: Measures on Schwarz distribution space and applications toP(φ)2 field theories. Adv. Math.12, 58–83 (1974)

    Google Scholar 

  43. Dobrushin, R.L.: The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Funct. Anal. Appl.2, 302–312 (1968)

    Google Scholar 

  44. Dobrushin, R.L., Minlos, R.A.: Construction of one dimensional quantum field via a continuous Markov field. Funct. Anal. Appl.7, 324–325 (1973)

    Google Scholar 

  45. Dobrushin, R.L., Minlos, R.A.: An investigation of the properties of generalized Gaussian random fields. Sel. Math. Sov.1, 215–263 (1981)

    Google Scholar 

  46. Dobrushin, R.L., Pecherski, E.A.: Uniqueness conditions for finitely dependent random fields. In: Random Fields (Esztergom, 1979). Fritz, J. et al. (eds). Amsterdam: North-Holland 1981

    Google Scholar 

  47. Dynkin, E.B.: Markov processes and random fields. Bull. Am. Math. Soc.3, 975–999 (1980)

    Google Scholar 

  48. Ekhaguere, G.O.S.: On notions of Markov property. J. Math. Phys.18, 2104–2107 (1977)

    Google Scholar 

  49. Ellis, R.S., Newman, C.: Limit theorems for sums of dependent random variables occuring in statistical mechanics. Z. Wahrscheinlichkeitstheor. Verw. Geb.44, 117–139 (1978)

    Google Scholar 

  50. Ellis, R.S., Newman, C.: Necessary and sufficient conditions for the GHS inequality with applications to analysis and probability. Trans. Am. Math. Soc.237, 83–99 (1978)

    Google Scholar 

  51. Föllmer, H.: On the global Markov property. In: Streit, L. (ed.). Quantum fields — algebras, processes, Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  52. Föllmer, H.: Phase transition and Martin boundary. Sém. Prob. IX, Lecture Notes in Mathematics. Vol. 465. Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  53. Föllmer, H.: Von der Brownschen Bewegung zum Brownschen Blatt: einige neuere Richtungen in der Theorie der stochastischen Prozesse. In: Perspective in mathematics. Basel: Birkhäuser 1984

    Google Scholar 

  54. Fortuin, C., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.72, 89–103 (1971)

    Google Scholar 

  55. Fröhlich, J.: Verification of axioms for Euclidean and relativistic fields and Haag's theorem in a class ofP(φ)2 models. Ann. Inst. H. Poincaré21, 271–317 (1974)

    Google Scholar 

  56. Fröhlich, J.: Schwinger functions and their generating functionals. I. Helv. Phys. Acta47, 265–306 (1974)

    Google Scholar 

  57. Fröhlich, J.: Schwinger functions and their generating functionals. II. Markovian and generalized path space measures onL 1. Adv. Math.23, 119–180 (1977)

    Google Scholar 

  58. Fröhlich, J., Park, Y.M.: Remarks on exponential interactions and the quantum sine-Gordon equation in two space-time dimensions. Helv. Phys. ActA50, 315–329 (1977)

    Google Scholar 

  59. Fröhlich, J., Seiler, E.: The massive Thirring-Schwinger model (QED)2: convergence of perturbation theory and particle structure. Helv. Phys. Acta49, 889–924 (1976)

    Google Scholar 

  60. Fröhlich, J., Simon, B.: Pure states for generalP(φ)2 theories: construction, regularity and variational equality. Ann. Math.105, 493–526 (1977)

    Google Scholar 

  61. Gelfand, I., Vilenkin, N.: Generalized functions, Vol. IV, (English translation). New York: Academic Press 1964

    Google Scholar 

  62. Georgii, H.O.: Gibbs measures and phase transitions. Berlin, New York: De Gruyter 1988

    Google Scholar 

  63. Gidas, B.: The Glimm-Jaffe-Spencer expansion for the classical boundary conditions and coexistence of phases in the λφ 42 Euclidean (quantum) field theory. Ann. Phys.118, 18–83 (1979)

    Google Scholar 

  64. Gielerak, R.: Verification of the global Markov property in some class of strongly coupled exponential interactions. J. Math. Phys.24, 347–355 (1983)

    Google Scholar 

  65. Gielerak, R.: Trigomometric perturbations of the Gaussian generalized fields. Dubna-Preprint (1986)

  66. Gielerak, R.: On the DLR equation for theP(ϕ)2 Euclidean quantum field theory. The uniqueness theorem, BiBoS-Preprint (1988)

  67. Gielerak, R., Zegarlinski, B.: Uniqueness and global almost Markov property for regularized Yukawa gases. Fortschr. Phys.32, 1–24 (1984)

    Google Scholar 

  68. Ginibre, J.: General formulation of Griffith's inequalities. Commun. Math. Phys.16, 310–328 (1970)

    Google Scholar 

  69. Glaser, V.: On the equivalence of the Euclidean and Wightman formulation of field theory. Commun. Math. Phys.37, 257–272 (1974)

    Google Scholar 

  70. Glimm, J., Jaffe, A.: Quantum physics, 2nd ed. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  71. Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(φ)2 model and other applications of high temperature expansions. Part I. Physics of quantum field models. In: Constructive quantum field theory. (Erice, 1973). Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  72. Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(φ)2 model and other applications of high temperature expansions. Part II. The cluster expansion. In: Constructive quantum field theory. (Erice, 1973). Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  73. Glimm, J., Jaffe, A., Spencer, T.: The Wightman axioms and particle structure in theP(φ)2 quantum field model. Ann. Math.100, 585–632 (1974)

    Google Scholar 

  74. Glimm, J., Jaffe, A., Spencer, T.: A convergent expansion about mean field theory. I. Ann. Phys.101, 610–630 (1976)

    Google Scholar 

  75. Glimm, J., Jaffe, A., Spencer, T.: A convergent expansion about mean field theory. II. Ann. Phys.101, 631–669 (1976)

    Google Scholar 

  76. Goldstein, S.: Remarks on the global Markov property. Commun. Math. Phys.74, 223–234 (1980)

    Google Scholar 

  77. Griffiths, R.B., Hurst, C., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys.11, 790–795 (1970)

    Google Scholar 

  78. Griffiths, R., Simon, B.: The (φ4)2 field theory as a classical Ising model. Commun. Math. Phys.33, 145–164 (1973)

    Google Scholar 

  79. Gross, L.: Analytic vectors for representations of the canonical commutation relations and nondegeneracy of ground states. J. Funct. Anal.17, 104–111 (1974)

    Google Scholar 

  80. Guerra, F.: Uniqueness of the vacuum energy density and van Hove phenomena in the infinite volume limit for two dimensional self-coupled Bose fields. Phys. Rev. Lett.28, 1213 (1972)

    Google Scholar 

  81. Guerra, F., Rosen, L., Simon, B.: TheP(φ)2 Euclidean quantum field theory as classical statistical mechanics. Ann. Math.101, 111–259 (1975)

    Google Scholar 

  82. Guerra, F., Rosen, L., Simon, B.: Boundary conditions in theP(φ)2 Euclidean field theory. Ann. Inst. H. Poincaré15, 213–334 (1976)

    Google Scholar 

  83. Hegerfeldt, G.C.: From Euclidean to relativistic fields and the notion of Markoff fields. Commun. Math. Phys.35, 155–171 (1974)

    Google Scholar 

  84. Heisenberg, W.: Pauli, W.: Zur Quantendynamik der Wellenfelder. Z. Phys.56, 1–61 (1929);59, 168–190 (1930)

    Google Scholar 

  85. Herbst, I.: On canonical quantum field theories. J. Math. Phys.17, 1210–1221 (1976)

    Google Scholar 

  86. Hida, T., Streit, L.: On quantum theory in terms of white noise. Nagoya Math. J.68, 21–34 (1977)

    Google Scholar 

  87. Higuchi, Y.: On the absence of non-translational by invariant Gibbs states for the two-dimensional Ising model. Proc. Esztergom. Fritz, J. et al. (eds.). Coll. Math. Soc. Bolyai, J. Amsterdam: North-Holland 1979

    Google Scholar 

  88. Holley, R., Stroock, D.W.: The DLR conditions for translation invariant Gaussian measures on 421-1(421-2). Z. Wahrscheinlichkeitstheor. Verw. Geb.53, 293–304 (1980)

    Google Scholar 

  89. Imbrie, J.: Mass spectrum of the two-dimensional λφ4−1/4φ2−μφ quantum field theory. Commun. Math. Phys.78, 169–200 (1980)

    Google Scholar 

  90. Imbrie, J.: Phase diagrams and cluster expansions for low temperatureP(φ)2 models. I. The phase diagram. II. The Schwinger functions. Commun. Math. Phys.82, 261–304, 305–343 (1981)

    Google Scholar 

  91. Israel, R.B.: Some examples concerning the global Markov property. Commun. Math. Phys.105, 669–673 (1986)

    Google Scholar 

  92. Kessler, Ch.: Attractiveness of interactions for binary lattice systems and the global Markov property. Stoch. Proc. Appl.24, 309–313 (1987)

    Google Scholar 

  93. Kessler, Ch.: Examples of extremal lattice fields without the global Markov property. Publ. RIMS21, 877–888 (1985)

    Google Scholar 

  94. Klein, A.: Gaussian OS-Positive Processes. Z. Wahrscheinlichkeitstheor. Verw. Geb40, 115–124 (1977)

    Google Scholar 

  95. Klein, A.: Stochastic processes associated with quantum systems. pp. 329–337. In: New stochastic methods in physics. De Witt-Morette, C., Elworthy, K.D. (eds.). Phys. Rep.77, (1981)

  96. Kolsrud, T.: On the Markov property for certain Gaussian random fields. Prob. Th. Rel. Fields74, 393–402 (1986)

    Google Scholar 

  97. Kotani, S.: On a Markov property for stationary Gaussian processes with a multidimensional parameter, pp. 239–250. In: Proc. 2nd Jap.-USSR Symp. Lecture Notes in Mathematics, Vol.330, Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  98. Kuik, R. On theq-state Potts model by means of noncommutative algebras. Groningen. Phys. Thesis (1986)

  99. Kusuoka, S.: Markov fields and local operators. J. Fac. Sci., Univ. Tokyo IA26, 199–212 (1979)

    Google Scholar 

  100. Kusuoka, S.: Dirichlet forms and diffusion processes on Banach spaces. J. Fac. Sci., Univ. Tokyo 1A29, 79–95 (1982)

    Google Scholar 

  101. Lanford, O., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys.13, 194–215 (1969)

    Google Scholar 

  102. Lebowitz, J.L., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. Math. Phys.25, 276–282 (1972)

    Google Scholar 

  103. Malyshev, V.A., Nikolaev, I.: Uniqueness of Gibbs fields via cluster expansions. J. Stat. Phys.35, 375–380 (1984)

    Google Scholar 

  104. Merlini, D.: Boundary conditions and cluster property in the two-dimensional Ising model. J. Stat. Phys.21, 6 (1979)

    Google Scholar 

  105. Molchan, G.M.: Characterization of Gaussian fields with Markovian property. Sov. Math. Dokl.12, 563–567 (1971) (transl.)

    Google Scholar 

  106. Nelson, E.: The construction of quantum fields from Markov fields. J. Funct. Anal.12, 97–112 (1973)

    Google Scholar 

  107. Nelson, E.: The free Markov field. J. Funct. Anal.12, 211–217 (1973)

    Google Scholar 

  108. Nelson, E.: Probability theory and euclidean field theory, pp. 94–104. In: Constructive quantum field theory. Velo, G., Wightman, A.S. (eds.). Lecture Notes in Physics Vol.25, Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  109. Nelson, E.: Quantum fields and Markoff fields. In: Partial differential equations. Spencer, D. (ed.). Symp. in Pure Math., Vol.23. A.M.S. Publications, pp. 413–420 (1973)

  110. Newman, C.: The construction of stationary two-dimensional Markoff fields with an application to quantum field theory. J. Funct. Anal.14, 44–61 (1973)

    Google Scholar 

  111. Newman, C.: Inequalities for Ising models and field theories which obey the Lee-Yang theorem. Commun. Math. Phys.41, 1–9 (1975)

    Google Scholar 

  112. Xank, N.X., Zessin, H.: Martin Dynkin boundary of mixed Poisson processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.37, 191–200 (1977)

    Google Scholar 

  113. Nualart, D., Zakai, N.: Generalized Brownian functionals and the solution to a stochastic partial differential equation. Barcelona-Haifa Preprint

  114. O'Carroll, M., Otterson, P.: Some aspects of Markov and Euclidean field theories. Commun. Math. Phys.36, 37–58 (1974)

    Google Scholar 

  115. Okabe, Y.: On the general fields of stationary Gaussian processes with Markovian properties. J. Math. Soc. Jpn.28, 86–95 (1976)

    Google Scholar 

  116. Osterwalder, K.: Euclidean Green's functions and Wightman distributions. In: Constructive quantum field theory. pp. 71–93. Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  117. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green's functions. I. Commun. Math. Phys.31, 83–112 (1973)

    Google Scholar 

  118. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green's functions. II. Commun. Math. Phys.42, 281–305 (1975)

    Google Scholar 

  119. Paclet, Ph.: Espaces de Dirichlet en dimension infinie. C. R. Ac. Sci., Paris, Ser. A288, 981–983 (1979)

    Google Scholar 

  120. Park, Y.M.: Massless quantum sine-Gordon equation in two space-time dimensions: Correlation inequalities and infinte volume limit. J. Math. Phys.18, 2423–2426 (1977)

    Google Scholar 

  121. Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems. Theor. Mat. Fiz.25, 358–369 [English transl.: Theor. Math. Phys.25, 1185–1192 (1975)

    Google Scholar 

  122. Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems, Continuation. Theor. Mat. Fiz.26, 61–76 [English transl.: Theor. Math. Phys.26, 39–49 (1976)

    Google Scholar 

  123. Pirogov, S.A., Sinai, Ya.G.: Ground states in two-dimensional boson quantum field-theory. Ann. Phys.109, 393–400 (1977)

    Google Scholar 

  124. Pitt, L.D.: A Markov property for Gaussian processes with a multidimensional parameter. Arch. Rat. Mech. Anal.43, 367–391 (1971)

    Google Scholar 

  125. Preston, C.: Random fields, Lecture Notes in Math. Vol.534. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  126. Röckner, M.: Markov property of generalized fields and axiomatic potential theory. Math. Ann.264, 153–177 (1983)

    Google Scholar 

  127. Röckner, M.: A Dirichlet problem for distributions and specifications for random fields. Mem. AMS54, 324 (1985)

    Google Scholar 

  128. Röckner, M.: Specifications and Martin boundary forP(φ)2 random fields. Commun. Math. Phys.106, 105–135 (1986)

    Google Scholar 

  129. Röckner, M., Zegarlinski, B.: The Dirichlet problem for quasilinear partial differential operators with boundary data given by a distribution. To appear in Proc. BiBoS Symp. III. Dortrecht: Reidel 1988

    Google Scholar 

  130. Rogers, L.C.G., Williams, D.: Diffusions, Markov processes, and martingales, Vol. 2 Ito calculus. Chichester: J. Wiley 1987

    Google Scholar 

  131. Royer, G.: Etude des champs euclidiens sur un réseauZ v. J. Math. Pures Appl.56, 455–478 (1977)

    Google Scholar 

  132. Rozanov, Yu.A.: Markov random fields. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  133. Rozanov, Yu.A.: Boundary problems for stochastic partial differential equations. In: Stochastic processes — Mathematics and physics. II. Alberverio, S., Blanchard, Ph., Streit, L. (eds.). Lecture Notes in Mathematics, vol. 1250, pp. 233–268. Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  134. Rozanov, Yu.A.: Some boundary value problems for generalized PDE, BiBoS-Preprint (1988)

  135. Russo, F.: A prediction problem for Gaussian planar processes which are Markovian with respect to increasing and decreasing paths. Lect. Notes Control Inf. Sciences96, 88–98 (1987)

    Google Scholar 

  136. Simon, B: TheP(φ)2 Euclidean (Quantum) field theory. Princeton, NJ: Princeton Univ. (1974)

    Google Scholar 

  137. Sinai, Ya.G.: The theory of phase transitions: rigorous results. London: Pergamon 1981

    Google Scholar 

  138. Spencer, T.: The mass gap for theP(φ)2 quantum field model with a strong external field. Commun. Math. Phys.39, 63–76 (1974)

    Google Scholar 

  139. Summers, S.J.: On the phase diagram of aP(φ)2 quantum field model. Ann. Inst. H. Poincaré34, 173–229 (1981)

    Google Scholar 

  140. Takeda, M.: On the uniqueness of Markov self-adjoint extensions of differential operators on infinite dimensional spaces. Osaka J. Math.22, 733–742 (1985)

    Google Scholar 

  141. Wick, W.D.: Convergence to equilibrium of the stochastic Heisenberg model. Math. Phys.81, 361–377 (1981)

    Google Scholar 

  142. Wong, E.: Homogenous Gauss-Markov random fields. Ann. Math. Stat.40, 1625–1634 (1969)

    Google Scholar 

  143. Zahradnik, M.: An alternative version of Pirogov-Sinai theory. Commun. Math. Phys.93, 559–581 (1984)

    Google Scholar 

  144. Zegarlinski, B.: Uniqueness and the global Markov property for Euclidean fields: the case of general exponential interaction. Commun. Math. Phys.96, 195–221 (1984)

    Google Scholar 

  145. Zegarlinski, B.: The Gibbs measures and partial differential equations. I. Commun. Math. Phys.107, 411–429 (1986)

    Google Scholar 

  146. Zegarlinski, B.: Extremality and the global Markov property. II. The global Markov property for non-FKG maximal measures. J. Stat. Phys.43, 687–705 (1986)

    Google Scholar 

  147. Zegarlinski, B.: The Gibbs measures and partial differential equations. II (unpublished)

  148. Zegarlinski, B.: Extremality and the global Markov property. I. The Euclidean fields on a lattice. J. Multiv. Anal.21, 158–167 (1987)

    Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität-Bochum, BiBoS-Bielefeld-Bochum, SFB 237, Bochum-Essen-Düsseldorf, Germany

    Sergio Albeverio

  2. Matematisk Institutt, Universitetet i Oslo, Blindern, Oslo 3, Norway

    Raphael Høegh-Krohn

  3. Institute of Theoretical Physics, University of Wrocław, Poland

    Boguslav Zegarlinski

  4. Theoretische Physik, ETH-Zürich, Hönggerberg, Switzerland

    Boguslav Zegarlinski

Authors
  1. Sergio Albeverio
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  2. Raphael Høegh-Krohn
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  3. Boguslav Zegarlinski
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Communicated by K. Gawedzki

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Albeverio, S., Høegh-Krohn, R. & Zegarlinski, B. Uniqueness and global Markov property for Euclidean fields: The case of general polynomial interactions. Commun.Math. Phys. 123, 377–424 (1989). https://doi.org/10.1007/BF01238808

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  • DOI: https://doi.org/10.1007/BF01238808

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