Abstract
The purpose of these lectures is to present some new developments in the theory of Dirichlet forms on infinite dimensional state space E. They are essentially based on joint work with Sergio Albeverio (cf. [AR 88a, b, 89a,b, 90] and also [AKR 88]) done during the last two to three years extending earlier fundamental work in [AH-K 75, 77a,b]. Section 5 is based on a very recent joint paper with Zhang Tu-Sheng (cf. [RZ 90]).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albeverio S., Brasche, J., Röckner, M., Dirichlet forms and generalized Schrödinger operators. In: Schrödinger Operators; eds. H. Holden, A. Jensen, Lecture Notes in Physics 345, Springer, 1989, pp. 1–42.
Albeverio, S., Hoegh-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.R.N.S., No. 248, Marseille, June 23–27, 1975, C.N.R.S., 1976.
Albeverio, S., Hoegh-Krohn, R., Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 40 (1977), 1–57.
Albeverio, S., Hoegh-Krohn, R., Hunt processes and analytic potential theory on rigged Hilbert spaces, Ann. Inst. Henri Poincaré Sect. B, 13 (1977), 269–291.
Albeverio, S., Hoegh-Krohn, R., Streit, L., Energy forms, Hamiltonians and distorted Brownian paths, J. Math. Phys. 18 (1977), 907–917.
Albeverio, S., Kusuoka, S., Röckner, M., On partial integration in infinite dimensional space and applications to Dirichlet forms, Preprint, 1988. J. London Math. Soc. 42 (1990), 122–136.
Albeverio, S., Röckner, M., New developments in theory and applications of Dirichlet forms. In: Stochastic Processes, Physics and Geometry, 27–76, Ascona/Locarno, Switzerland, July 4–9, 1988. Eds.: S. Albeverio et al, Singapore World Scientific, 1990.
Albeverio, S., Röckner, M., On the maximality problem for classical Dinchlet forms on topological vector spaces, Proc. Conference Bad-Honnef, June 1988, eds. N. Christopeit, K. Helmes, M. Kohlmann, Lecture Notes Inform. Control, 126, Springer, 1989, pp. 14–31.
Albeverio, S., Röckner, M., Classical Dirichlet forms on topological vector spaces—construction of an associated diffusion process, Prob. Th. Rel. Fields, 83, (1989), 405–434.
Albeverio, S., Röckner, M., Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Preprint, Edinburgh, 1989. Prob. Th. Rel. Fields, 89 (1991), 347–386.
Albeverio, S., Röckner, M., Classical Dirichlet forms on topological vector spaces—closability and a Cameron-Martin formula, J. Funct. Anal. 88 (1990), 395–436.
Borkar, B.S., Chari, R.T., Mitter, S.K., Stochastic quantization of field theory infinite and infinite volume, J. Funct. Anal. 81, (1988), 184–206.
Bouleau, N., Hirsch, F., Formes de Dirichlet générales et densité des variables aléatoires réelles sur l’espace de Wiener, J. Funct. Anal. 69 (1986), 229–259.
Dynkin, E.B., Markov Processes, Vols. I and II, Berlin-Heidelberg-New York: Springer, 1965.
Dynkin, E.B., Integral representation of excessive measures and excessive functions, Uspehi Mat. Nauk 27, Vol. 1, (1972), 43–90. English translation: Russian Math. Surveys 27, Vol. 1, (1972), 43–84.
Engelbert, H.J., Schmidt, W., On one-dimensional stochastic differential equations with generalized drift, Lectures Notes in Control and Information Sciences 69, Berlin: Springer, 1984, pp. 143–155.
Fukushima, M., Dirichlet Forms and Markov Processes, Amsterdam-Oxford-New York: North Holland, 1980.
Fukushima, M., On a stochastic calculus related to Dirichlet forms and distorted Brownian motion, Physical Reports, 77 (1981), 255–262.
Fukushima, M., On absolute continuity of multidimensional sym-metrizable diffusions. In: Lecture Notes in Math. 923, Berlin-Heidelberg-New York: Springer, 1982, pp. 146–176.
Fukushima, M., Energy forms and diffusion process. In: Mathematics and Physics, Lectures on Recent Results, Ed. Streit, L., Singapore: World Scientific Publishing Co., 1984.
Gamelin, T.W., Uniform Algebras, Englewood Cliffs: Prentice Hall, 1969.
Glimm, J., Jaffe, A., Quantum Physics: A Functional Integral Point of View, New York-Heidelberg-Berlin: Springer, 1986.
Gross, L., Abstract Wiener Spaces, Proc. 5th Berkeley Symp. Math. Stat. Prob. 2, (1965), 31–42.
Hamza, M.M., Détermination des forms de Dirichlet sur ∝ n. Thèse Seme Cycle, Orsay, 1975.
Ito, K., Infinite dimensional Ornstein-Uhlenbeck processes. In: Stochastic Analysis, Ed. K. Ito, Amsterdam-Oxford-New York: North Holland, 1984, pp. 197–224.
Jona-Lasinio, P., Mitter, P.K., On the stochastic quantization of field theory, Comm. Math. Phys. 101 (1985), 409–436.
Kreé, P., Calcul d’integrales et de dérivées en dimension infinie, J. Funct. Anal. 31 (1979), 150–186.
Kuo, H., Gaussian measures in Banach spaces, Lecture Notes in Math. 463, Berlin-Heidelberg-New York: Springer, 1975, pp. 1–224.
Kusuoka, S., Dirichlet forms and diffusion processes on Banach spaces, J. Fac. Sci. Univ. Tokyo Sect. IA 29 (1982), 79–85.
Lions, J.L., Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Grundlehren Math. Wiss., Berlin: Springer, 1972.
Lyons, T., Röckner, M., A note on tightness of capacities associated with Dirichlet forms, preprint Edinburgh, 1990. Bull London Math. Soc. 24 (1992), 181–184.
Malliavin, P., Stochastic calculus of variation and hypoelliptic operators, Proc. of the International Symposium on Stochastic Differential Equations, Kyoto 1976 Tokyo 1978.
Mizohata, S., The Theory of Partial Differential Equations, London: Cambridge University Press, 1973.
Parthasarathy, K.R., Probability Measures on Metric Spaces, New York-London: Academic Press, 1967.
Pedersen, G.K., Analysis Now, New York-Heidelberg-London-Paris-Tokyo: Springer, 1988.
Potthoff, J., Röckner, M., On the contraction property of infinite dimensional Dirichlet forms, Preprint, Edinburgh 1989. J. Funct. Anal. 92 (1990), 155–165.
Reed, M., Simon, B., Methods of Modern Mathematical Physics IV. Analysis of Operators, New York-San Francisco-London: Academic Press, 1978.
Röckner, M., Specifications and Martin boundaries for P (Φ)2-ran-dom fields, Comm. Math. Phys. 106 (1986), 105–135.
Röckner, M., Zhang, T., On uniqueness of generalized Schrödinger operators and applications, Preprint Edinburgh (1990). J. Funct. Anal. 105 (1992), 187–231.
Rullkötter, K., Spönemann, U., Dirichletformen und Diffusionsprozesse, Diplomarbeit, Bielefeld, 1983.
Schmuland, B., An alternative compactification for classical Dirichlet forms on topological vector spaces, Preprint 1989. Stochastics 33 (1990), 75–90.
Schwartz, L., Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, London: Oxford University Press, 1973.
Shigekawa, I., Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator, Preprint 1990.
Silverstein, M.L., Symmetric Markov Processes, Lecture Notes in Math. 426, Berlin-Heidelberg-New York: Springer, 1974.
Simon, B., The P(Φ) 2 Euclidean (Quantum) Field Theory, Princeton University Press, 1974.
Spönemann, U., Ph.D. Thesis, Bielefeld. Publication in preparation.
Sugita, H., Sobolev spaces of Wiener functionals and Malliavin’s calculus, J. Math. Kyoto Univ. 25 (1985), 31–48.
Sugita, H., On a characterization of the Sobolev spaces over an abstract Wiener space, J. Math. Kyoto Univ. 25 (1985), 717–725.
Takeda, M., On the uniqueness of Markovian self-adjoint extension of diffusion operators on infinite dimensional space, Osaka J. Math. 22 (1985), 733–742.
Takeda, M., On the uniqueness of the Markovian self-adjoint extension, Lecture Notes in Math. 1250 (Stochastic Processes — Mathematics and Physics), 1985, pp. 319–325.
Takeda, M., The maximum Markovian self-adjoint extensions of generalized Schrödinger operators, Preprint (1990). J. Math. Soc. Japan 44 (1992), 113–130.
Watanabe, S., Lectures on Stochastic Differential Equations and Malliavin Calculus, Berlin-Heidelberg-New York-Tokyo: Springer, 1984.
Wielens, N., On the essential self-adjointness of generalized Schrödinger operators, J. Funct. Anal. 61 (1985), 98–115.
Yan, J.A., Generalizations of Gross’ and Minlos’ theorems. In: Séminaire de Probabilités XXII, eds. J. Azema, P.A. Meyer, M. Yor. Lect. Notes in Math. 1372, Springer, 1989, pp. 395–404.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media New York
About this chapter
Cite this chapter
Röckner, M. (1992). Dirichlet Forms on Infinite Dimensional State Space and Applications. In: Körezlioğlu, H., Üstünel, A.S. (eds) Stochastic Analysis and Related Topics. Progress in Probability, vol 31. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0373-5_2
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0373-5_2
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6731-7
Online ISBN: 978-1-4612-0373-5
eBook Packages: Springer Book Archive