Classical Dirichlet forms on topological vector spaces - the construction of the associated diffusion process. (English) Zbl 0661.60094
Given a (minimal) classical Dirichlet form on \(L^ 2(E;\mu)\) we construct the associated diffusion process. Here E is a locally convex topological vector space and \(\mu\) is a (not necessarily quasi-invariant) probability measure on E. The construction is carried out under certain assumptions on E and \(\mu\) 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 without cut-off).
Reviewer: S.Albeverio
MSC:
| 60J60 | Diffusion processes |
| 31C25 | Dirichlet forms |
| 60J45 | Probabilistic potential theory |
| 60G60 | Random fields |
Keywords:
Dirichlet form; quantum fields; infinite dimensional diffusion processes; infinite dimensional calculus; capacities on Souslin spaces; disintegration of measuresReferences:
| [1] | Albeverio, S.; Christopeit, N.; Helmes, K.; Kohlman, M., Some points of interaction between stochastic analysis and quantum theory, Stochastic differential systems. Proceedings, Bad Honnef 1985, 1-26 (1986), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0601.60069 |
| [2] | Albeverio, S.; Fenstad, J. E.; Høegh-Krohn, R.; Lindstrøm, T., Nonstandard methods in stochastic analysis and mathematical physics (1986), New York London: Academic Press, New York London · Zbl 0605.60005 |
| [3] | Albeverio, S., Hida, T., Potthoff, J., Röckner, M., Streit, L.: Dirichlet forms in terms of white noise analysis I+II. BiBoS-preprints (1989) · Zbl 0708.60100 |
| [4] | 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) · Zbl 0279.60095 |
| [5] | 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 |
| [6] | Albeverio, S.; Høegh-Krohn, R., Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z. Wahrscheinlichkeitstheor. Verw. Geb., 40, 1-57 (1977) · Zbl 0342.60057 |
| [7] | Albeverio, S.; Høegh-Krohn, R., Hunt processes and analytic potential theory on rigged Hilbert spaces, Ann. Inst. Henri Poincaré, 8, 269-291 (1977) · Zbl 0382.60082 |
| [8] | 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) · Zbl 0443.60099 |
| [9] | Albeverio, S.; Høegh-Krohn, R.; Pinsky, M. A., Diffusion fields, quantum fields, and fields with values in Lie groups, Stochastic analysis and applications (1984), New York: Marcel Dekker, New York · Zbl 0575.60078 |
| [10] | 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 · Zbl 0798.46055 |
| [11] | Albeverio, S., Röckner, M.: Classical Dirichlet forms on topological vector spaces — closability and a Cameron-Martin formula. J. Funct. Anal. (1989) · Zbl 0737.46036 |
| [12] | Albeverio, S.; Röckner, M.; Elworthy, R. D.; Zambrini, J. C., Dirichlet forms, quantum fields and stochastic quantization, Stochastic analysis, path integration and dynamics, 1-21 (1989), Harlow: Longman, Harlow · Zbl 0691.60044 |
| [13] | Badrikian, A., Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques (1970), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0209.48402 |
| [14] | Bauer, H., Wahrscheinlichkeitstheorie und Grundzüge der Maßtheorie (1978), Berlin New York: de Gruyter, Berlin New York · Zbl 0381.60001 |
| [15] | 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) · Zbl 0657.60084 |
| [16] | 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) · Zbl 0605.60058 |
| [17] | Bourbaki, N., Topologie générale (1974), Paris: Hermann, Paris · Zbl 0337.54001 |
| [18] | Dellacherie, C.; Meyer, P. A., Probabilities and potential (1978), Amsterdam New York Oxford: North-Holland, Amsterdam New York Oxford · Zbl 0494.60001 |
| [19] | Dixmier, J., Les algèbres d’opérateurs dans l’espace hilbertien (1969), Paris: Gauthier-Villars, Paris · Zbl 0175.43801 |
| [20] | Dobrushin, R. I.; Minlos, R. A., The moments and polynomials of a generalized random field, Theor. Probab. Appl., 23, 686-699 (1978) |
| [21] | Döring, C. R., Nonlinear parabolic stochastic differential equations with additive coloured noise on ℝ^d×ℝ_+: a regulated stochastic quantization, Commun. Math. Phys., 109, 537-561 (1987) · Zbl 0625.60069 |
| [22] | Dynkin, E. B., Markov processes vols. I and II (1965), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0132.37901 |
| [23] | Dynkin, E. B., Green’s and Dirichlet spaces associated with fine Markov processes, J. Funct. Anal., 47, 381-418 (1982) · Zbl 0488.60083 |
| [24] | Dynkin, E.B.: Green’s and Dirichlet spaces for a symmetric Markov transition function. (Preprint 1982) |
| [25] | Föllmer, H., Phase transition and Martin boundary, Séminaire de probabilités IX, Strasbourg (1975), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0367.60112 |
| [26] | Fröhlich, J., Schwinger functions and their generating functionals, I, Helv. Phys. Acta, 47, 265-306 (1974) · Zbl 0345.46057 |
| [27] | Fröhlich, J., Schwinger functions and their generating functionals, II. Markovian and generalized path space measures onL, Adv. Math., 23, 119-180 (1977) · Zbl 0345.46058 |
| [28] | 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) |
| [29] | Fröhlich, J.; Simon, B., Pure states for generalP(Φ)_2-theories: construction, regularity and variational equality, Ann. Math., 105, 493-526 (1977) |
| [30] | Fukushima, M., Regular representations of Dirichlet forms, Trans. Amer. Math. Soc., 155, 455-473 (1971) · Zbl 0248.31007 |
| [31] | Fukushima, M., Dirichlet spaces and strong Markov processes, Trans. Amer. Math. Soc., 162, 185-224 (1971) · Zbl 0254.60055 |
| [32] | Fukushima, M., Dirichlet forms and Markov processes (1980), Amsterdam Oxford New York: North-Holland, Amsterdam Oxford New York · Zbl 0422.31007 |
| [33] | Fukushima, M., Basic properties of Brownian motion and a capacity on the Wiener space, J. Math. Soc. Japan, 36, 161-175 (1984) · Zbl 0522.60081 |
| [34] | Gamelin, T. W., Uniform algebras (1969), Englewood Cliffs: Prentice Hall, Englewood Cliffs · Zbl 0213.40401 |
| [35] | Gelfand, I. M.; Vilenkin, N. J., Generalized functions, vol. 4. Some applications of harmonic analysis (1964), New York: Academic Press, New York · Zbl 0136.11201 |
| [36] | Getoor, R. K., Markov processes: Ray processes and right processes (1975), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0299.60051 |
| [37] | Glimm, J.; Jaffe, A., Entropy principle for vertex functions in quantum field models, Ann. Inst. Henri Poincaré, 21, 1-26 (1974) |
| [38] | Glimm, J.; Jaffe, A., Quantum physics: A functional integral point of view (1981), New York Heidelberg Berlin: Springer, New York Heidelberg Berlin · Zbl 0461.46051 |
| [39] | Glimm, J.; Jaffe, A.; Spencer, T.; Velo, G.; Wightman, A., The particle structure of the weakly coupledP(Φ)_2 model and other applications of high temperature expansions, Constructive quantum field theory (1973), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York |
| [40] | 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) |
| [41] | Gross, L., Abstract Wiener spaces, Proc. 5th Berkeley Symp. Math. Stat. Prob., 2, 31-42 (1965) · Zbl 0187.40903 |
| [42] | Guerra, F.; Rosen, J.; Simon, B., TheP(Φ)_2 Euclideam quantum field theory as classical statistical mechanics, Ann. Math., 101, 111-259 (1975) · Zbl 1495.82015 |
| [43] | Guerra, F.; Rosen, J.; Simon, B., Boundary conditions in theP(Φ)_2 Euclidean field theory, Ann. Inst. Henri Poincaré, 15, 231-334 (1976) · Zbl 1495.81067 |
| [44] | Haba, Z., Some non-Markovian Osterwalder-Schrader fields, Ann. Inst. Henri Poincaré, Sect. A (N.S.), 32, 185-201 (1980) · Zbl 0427.60097 |
| [45] | Hamza, M.M.: Détermination des formes de Dirichlet sur ℝ^n. Thèse 3eme cycle, Orsay (1975) |
| [46] | Hida, T., Brownian motion (1980), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0432.60002 |
| [47] | Jona-Lasinio, P.; Mitter, P. K., On the stochastic quantization of field theory, Commun. Math. Phys., 101, 409-436 (1985) · Zbl 0588.60054 |
| [48] | Kuo, H., Gaussian measures in Banach spaces, 1-224 (1975), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0306.28010 |
| [49] | Kusuoka, S., Dirichlet forms and diffusion processes on Banach space, J. Fac. Science Univ. Tokyo, Sec. 1 A, 29, 79-95 (1982) · Zbl 0496.60079 |
| [50] | Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators. Proc. of the International Symposium on Stochastic Differential Equations, Kyoto 1976, Tokyo 1978 · Zbl 0411.60060 |
| [51] | Mitter, P. K.; Abad, J.; Asorey, M.; Cruz, A., Stochastic approach to Euclidean field theory (Stochastic Quantization), New perspectives in quantum field theories (1986), Singapore: World Scientific, Singapore |
| [52] | Nelson, E., The free Markov field, J. Funct. Anal., 12, 221-227 (1973) · Zbl 0273.60079 |
| [53] | Parisi, G.; Wu, Y. S., Perturbation theory without gauge fixing, Sci. Sin., 24, 483-496 (1981) · Zbl 1480.81051 |
| [54] | Parthasathy, K. R., Probability measures on metric spaces (1967), New York London: Academic Press, New York London · Zbl 0153.19101 |
| [55] | Potthoff, J., Röckner, M.: On the contraction property of Dirichlet forms on infinite dimensional space. Preprint, Edinburgh, 1989, to appear in J. Funct. Anal. · Zbl 0708.46045 |
| [56] | Preston, C., Random fields (1976), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0335.60074 |
| [57] | Reed, M.; Simon, B., Methods of modern mathematical physics. I. Functional analysis (1972), New York London: Academic Press, New York London · Zbl 0242.46001 |
| [58] | Ripley, B. D., The disintegration of invariant measures, Math. Proc. Camb. Phil. Soc., 79, 337-341 (1976) · Zbl 0316.28006 |
| [59] | Röckner, M., Generalized Markov fields and Dirichlet forms, Acta Appl. Math., 3, 285-311 (1985) · Zbl 0588.60044 |
| [60] | Röckner, M., Specifications and Martin boundaries forP(Φ)_2-random fields, Commun. Math. Phys., 106, 105-135 (1986) · Zbl 0614.60068 |
| [61] | Röckner, M., Traces of harmonic functions and a new path space for the free quantum field, J. Funct. Anal., 79, 211-249 (1988) · Zbl 0642.60055 |
| [62] | Röckner, M.; Král, J.; Lukeš, J.; Netoka, I.; Veselý, J., On the transition function of the infinite dimensional Ornstein-Uhlenbeck process given by the free quantum field, Potential theory (1988), New York London: Plenum Press, New York London · Zbl 0686.60076 |
| [63] | Röckner, M.; Wielens, N.; Albeverio, S., Dirichlet forms — closability and change of speed measure, Infinite dimensional analysis and stochastic processes (1985), Boston London Melbourne: Pitman, Boston London Melbourne · Zbl 0573.31005 |
| [64] | Rullkötter, K., Spönemann, U: Dirichletformen und Diffusionsprozesse. Diplomarbeit, Bielefeld (1983) |
| [65] | Schwartz, L., Radon measures on arbitrary topological spacas and cylindrical measures (1973), London: Oxford University Press, London · Zbl 0298.28001 |
| [66] | Silverstein, M. L., Symmetric Markov processes (1974), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0296.60038 |
| [67] | Simon, B., TheP(Φ)_2 Euclidean (quantum) field theory (1974), Princeton: Princeton University Press, Princeton · Zbl 1175.81146 |
| [68] | Spönemann, U.: PhD thesis, Bielefeld 1989. |
| [69] | Steffens, J.: Excessive measures and the existence of right semigroups and processes. Preprint · Zbl 0658.60108 |
| [70] | Takesaki, M., Theory of operator algebras I (1979), New York Heidelberg Berlin: Springer, New York Heidelberg Berlin · Zbl 0436.46043 |
| [71] | Watanabe, S., Lectures on stochastic differential equations and Malliavin calculus (1984), Berlin Heidelberg New York Tokyo: Springer, Berlin Heidelberg New York Tokyo · Zbl 0546.60054 |
| [72] | Zegarlinski, B., Uniqueness and the global Markov property for Euclidean fields: The case of general exponential interaction, Commun. Math. Phys., 96, 195-221 (1984) · Zbl 0576.60097 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
