Summary
Given a (minimal) classical Dirichlet form onL 2 (E;μ) we construct the associated diffusion process. HereE is a locally convex topological vector space and μ is a (not necessarily quasi-invariant) probability measure onE. The construction is carried out under certain assumptions onE and μ which can be easily verified in many examples. In particular, we explicitly apply our results to (time-zero and space-time) quantum fields (with or with-out cut-off).
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S.Albeverio: Some points of interaction between stochastic analysis and quantum theory. In: Christopeit, N., Helmes, K., Kohlman, M. (eds.) Stochastic differential systems. Proceedings, Bad Honnef 1985. (Lect. Notes Contr. Inf. Sci., vol. 18, pp. 1–26) Berlin Heidelberg New York: Springer 1986
Albeverio, S., Fenstad, J.E., Høegh-Krohn, R., Lindstrøm, T.: Nonstandard methods in stochastic analysis and mathematical physics. New York London: Academic Press 1986
Albeverio, S., Hida, T., Potthoff, J., Röckner, M., Streit, L.: Dirichlet forms in terms of white noise analysis I+II. BiBoS-preprints (1989)
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)
Albeverio, S., Høegh-Krohn R.: Quasi-invariant measures, symmetric diffusion processes and quantum fields. In. Les méthodes mathématiques de la théorie quantique des champs, Colloques Internationaux du C.N.R.S., no. 248, Marseille, 23–27 juin 1975, C.N.R.S., 1976
Albeverio, S., Høegh-Krohn, R.: Dirichlet forms and diffusion processes on rigged Hilbert spaces. Z. Wahrscheinlichkeitstheor. Verw. Geb.40, 1–57 (1977)
Albeverio, S., Høegh-Krohn, R.: Hunt processes and analytic potential theory on rigged Hilbert spaces. Ann. Inst. Henri Poincaré8, 269–291 (1977)
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)
Albeverio, S., Høegh-Krohn, R.: Diffusion fields, quantum fields, and fields with values in Lie groups. In: Pinsky, M.A. (ed.) Stochastic analysis and applications. New York: Marcel Dekker 1984
Albeverio, S., Kusuoka, S.: Maximality of infinite dimensional Dirichlet forms and Høegh-Krohn's model of quantum fields. Kyoto-Bochum Preprint (1988), to appear in Mem. Volume for R. Høegh-Krohn
Albeverio, S., Röckner, M.: Classical Dirichlet forms on topological vector spaces — closability and a Cameron-Martin formula. J. Funct. Anal. (1989)
Albeverio, S., Röckner, M.: Dirichlet forms, quantum fields and stochastic quantization. In: Elworthy, R.D., Zambrini, J.C. (eds.) Stochastic analysis, path integration and dynamics. (Pitman Res. Notes, vol. 200, pp. 1–21) Harlow Longman 1989
Badrikian, A.: Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques. (Lect. Notes Math., vol. 139) Berlin Heidelberg New York: Springer 1970
Bauer, H.: Wahrscheinlichkeitstheorie und Grundzüge der Maßtheorie. Berlin New York: de Gruyter 1978
Borkar, V.S., Chari, R.T., Mitter, S.K.: Stochastic quantization of field theory in finite and infinite volume. J. Funct. Anal.81, 184–206 (1988)
Bouleau, N., Hirsch, F.: Formes de Dirichlet générales et densité des variables aléatoires réelles sur l'espace de Wiener. J. Funct. Anal.69, 229–259 (1986)
Bourbaki, N.: Topologie générale, Chapitres 5 à 10. Paris: Hermann 1974
Dellacherie, C., Meyer, P.A.: Probabilities and potential. Amsterdam New York Oxford: North-Holland 1978
Dixmier, J.: Les algèbres d'opérateurs dans l'espace hilbertien. Paris: Gauthier-Villars 1969
Dobrushin, R.I., Minlos, R.A.: The moments and polynomials of a generalized random field. Theor. Probab. Appl.23, 686–699 (1978)
Döring, C.R.: Nonlinear parabolic stochastic differential equations with additive coloured noise on 432-1: a regulated stochastic quantization. Commun. Math. Phys.109, 537–561 (1987)
Dynkin, E.B.: Markov processes vols. I and II. Berlin Heidelberg New York: Springer 1965
Dynkin, E.B.: Green's and Dirichlet spaces associated with fine Markov processes. J. Funct. Anal.47, 381–418 (1982)
Dynkin, E.B.: Green's and Dirichlet spaces for a symmetric Markov transition function. (Preprint 1982)
Föllmer, H.: Phase transition and Martin boundary. Séminaire de probabilités IX, Strasbourg. (Lect. Notes Math., vol. 465) Berlin Heidelberg New York: Springer 1975
Fröhlich, J.: Schwinger functions and their generating functionals, I. Helv. Phys. Acta47, 265–306 (1974)
Fröhlich, J.: Schwinger functions and their generating functionals, II. Markovian and generalized path space measures onL. Adv. Math.23, 119–180 (1977)
Fröhlich, J., Israel, R., Lieb, E., Simon, B.: Phase transitions and reflection positivity. I. General theory and long-range lattice models. Commun. Math. Phys.62, 1–34 (1978)
Fröhlich, J., Simon, B.: Pure states for general 432-3: construction, regularity and variational equality. Ann. Math.105, 493–526 (1977)
Fukushima, M.: Regular representations of Dirichlet forms. Trans. Amer. Math. Soc.155, 455–473 (1971)
Fukushima, M.: Dirichlet spaces and strong Markov processes. Trans. Amer. Math. Soc.162, 185–224 (1971)
Fukushima, M.: Dirichlet forms and Markov processes. Amsterdam Oxford New York. North-Holland 1980
Fukushima, M.: Basic properties of Brownian motion and a capacity on the Wiener space. J. Math. Soc. Japan36, 161–175 (1984)
Gamelin, T.W.: Uniform algebras, Englewood Cliffs: Prentice Hall 1969
Gelfand, I.M., Vilenkin, N.J.: Generalized functions, vol. 4. Some applications of harmonic analysis. New York: Academic Press 1964
Getoor, R.K.: Markov processes: Ray processes and right processes. (Lect. Notes Math., vol. 440) Berlin Heidelberg New York: Springer 1975
Glimm, J., Jaffe, A.: Entropy principle for vertex functions in quantum field models. Ann. Inst. Henri Poincaré21, 1–26 (1974)
Glimm, J., Jaffe, A.: Quantum physics: A functional integral point of view. New York Heidelberg Berlin: Springer 1981
Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupled 433-1 model and other applications of high temperature expansions. In: Velo, G., Wightman, A. (eds.) Constructive quantum field theory. Berlin Heidelberg New York; Springer 1973
Glimm, J., Jaffe, A., Spencer, T.: The Wightman axioms and particle structure in the 433-2 quantum field model. Ann. Math.100, 585–632 (1974)
Gross, L.: Abstract Wiener spaces. Proc. 5th Berkeley Symp. Math. Stat. Prob.2, 31–42 (1965)
Guerra, F., Rosen, J., Simon, B.: The 433-3 Euclidean quantum field theory as classical statistical mechanics. Ann. Math.101, 111–259 (1975)
Guerra, F., Rosen, J., Simon, B.: Boundary conditions in the 433-4 Euclidean field theory. Ann. Inst. Henri Poincaré15, 231–334 (1976)
Haba, Z.: Some non-Markovian Osterwalder-Schrader fields, Ann. Inst. Henri Poincaré, Sect. A (N.S.)32, 185–201 (1980)
Hamza, M.M.: Détermination des formes de Dirichlet sur ℝn. Thèse 3eme cycle, Orsay (1975)
Hida, T.: Brownian motion. Berlin Heidelberg New York: Springer 1980
Jona-Lasinio, P., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys.101, 409–436 (1985)
Kuo, H.: Gaussian measures in Banach spaces. (Lect. Notes Math., vol. 463, pp. 1–224) Berlin Heidelberg New York; Springer 1975
Kusuoka, S.: Dirichlet forms and diffusion processes on Banach space. J. Fac. Science Univ. Tokyo, Sec. 1 A29, 79–95 (1982)
Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators. Proc. of the International Symposium on Stochastic Differential Equations, Kyoto 1976, Tokyo 1978
Mitter, P.K.: Stochastic approach to Euclidean field theory (Stochastic Quantization). In: Abad, J., Asorey, M., Cruz, A. (eds.) New perspectives in quantum field theories. Singapore: World Scientific 1986
Nelson, E.: The free Markov field. J. Funct. Anal.12, 221–227 (1973)
[Pa/Wu]Parisi, G., Wu, Y.S.: Perturbation theory without gauge fixing. Sci. Sin.24,483–496 (1981)
Parthasathy, K.R.: Probability measures on metric spaces. New York London: Academic Press 1967
Potthoff, J., Röckner, M.: On the contraction property of Dirichlet forms on infinite dimensional space. Preprint, Edinburgh, 1989, to appear in J. Funct. Anal.
Preston, C.: Random fields. (Lect. Notes Math., vol. 534) Berlin Heidelberg New York: Springer 1976
Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. New York London: Academic Press 1972
Ripley, B.D.: The disintegration of invariant measures. Math. Proc. Camb. Phil. Soc.79, 337–341 (1976)
Röckner, M.: Generalized Markov fields and Dirichlet forms. Acta Appl. Math.3, 285–311 (1985)
Röckner, M.: Specifications and Martin boundaries for 433-6 fields. Commun. Math. Phys.106,105–135 (1986)
Röckner, M.: Traces of harmonic functions and a new path space for the free quantum field. J. Funct. Anal.79, 211–249 (1988)
Röckner, M.: On the transition function of the infinite dimensional Ornstein-Uhlenbeck process given by the free quantum field. In: Král, J., Lukeš, J., Netoka, I., Veselý, J. (eds.) Potential theory. New York London: Plenum Press 1988
Röckner, M., Wielens, N.: Dirichlet forms — closability and change of speed measure. In: Albeverio, S. (ed.) Infinite dimensional analysis and stochastic processes. Boston London Melbourne: Pitman 1985
Rullkötter, K., Spönemann, U: Dirichletformen und Diffusionsprozesse. Diplomarbeit, Bielefeld (1983)
Schwartz, L.: Radon measures on arbitrary topological spacas and cylindrical measures. London: Oxford University Press 1973
Silverstein, M.L.: Symmetric Markov processes. (Lect. Notes Math., vol. 426) Berlin Heidelberg New York: Springer 1974
Simon, B.: The 434-1 Euclidean (quantum) field theory. Princeton: Princeton University Press 1974
Spönemann, U.: PhD thesis, Bielefeld 1989.
Steffens, J.: Excessive measures and the existence of right semigroups and processes. Preprint
Takesaki, M.: Theory of operator algebras I. New York Heidelberg Berlin: Springer 1979
Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Berlin Heidelberg New York Tokyo: Springer 1984
Zegarlinski, B.: Uniqueness and the global Markov property for Euclidean fields: The case of general exponential interaction. Commun. Math. Phys.96,195–221 (1984)
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Dedication: Raphael Høegh-Krohn (1938–1988) was an initiator of the theory of Dirichlet forms over infinite dimensional spaces. He has been a continuous source of inspiration. We deeply mourn his departure and dedicate to him this work, as a small sign of our great gratitude
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Albeverio, S., Röckner, M. Classical dirichlet forms on topological vector spaces-the construction of the associated diffusion process. Probab. Th. Rel. Fields 83, 405–434 (1989). https://doi.org/10.1007/BF00964372
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DOI: https://doi.org/10.1007/BF00964372