Skip to main content
SpringerLink
Log in

Markov Processes on Infinite Dimensional Spaces, Markov Fields and Markov Cosurfaces

  • Chapter
Stochastic Space—Time Models and Limit Theorems

Part of the book series: Mathematics and Its Applications ((MAIA,volume 19))

Abstract

We review work on Dirichlet forms and symmetric Markov processes on infinite dimensional spaces. Especially we consider the connections with the construction of homogeneous generalized Markov random fields. We also discuss a non commutative extension to the case where the state space is a group. The extension involves a stochastic calculus for group valued mappings defined on hypersurfaces of codimension 1.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Fukushima, Dirichlet forms and Markov processes, North Holland Kodansha (1980)

    MATH  Google Scholar 

  2. M.L. Silverstein, Symmetric Markov processes, Lect. Notes Maths. 426, Springer, Berlin (1974)

    MATH  Google Scholar 

  3. M.L. Silverstein, Boundary theory for symmetric Markov processes, Lect. Notes Maths. 516, Springer, Berlin (1976)

    MATH  Google Scholar 

  4. M. Fukushima, Energy forms and diffusion processes, ZiF-Preprint, in preparation (to appear in Proc. Bielefeld, Project 2: Mathematics — Physics, Ed. L. Streit, World Publishing, Singapore)

    Google Scholar 

  5. E. Nelson, Dynamical Theories of Brownian motion, Princeton University Press, 1970

    Google Scholar 

  6. E. Nelson Quantum Fluctuations, Princeton University, in preparation

    Google Scholar 

  7. S. Albeverio, R. Høegh-Krohn, A remark on the connection between stochastic mechanics and the heat equation, J. Math. Phys. 15, 1745–1747 (1974)

    Article  MATH  Google Scholar 

  8. S. Albeverio, R. Høegh-Krohn, 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”, Ed. CNRS (1976)

    Google Scholar 

  9. S. Albeverio, R. Høegh-Krohn, L. Streit, Energy forms, Hamiltonians and distorted paths, J. Math. Phys. 18, 907–917 (1977)

    Article  MATH  Google Scholar 

  10. M. Fukushima, A decomposition of additive functionals of finite energy, Nagoya Math. J. 74, 137–168 (1979)

    MathSciNet  MATH  Google Scholar 

  11. M. Fukushima, On a representation of local martingale additive functionals of symmetric diffusions, pp. 110–118 in Stochastic Integrals, Ed. D. Williams, Lect. Notes Maths. 851, Springer, Berlin (1981)

    Chapter  Google Scholar 

  12. M. Röckner, N. Wielens, Dirichlet forms — closability and change of speed measure, ZiF-Preprint, to appear in “Infinite dimensional analysis and stochastic processes”, Ed. S. Albeverio, Res. Notes Maths., Pitman

    Google Scholar 

  13. K. Rullkötter, U. Spönemann, Dirichletformen und Dif-fussionsprozesse, Diplomarbeit, Bielefeld; and paper in preparation

    Google Scholar 

  14. J. Brasche, Störungen von Schrödingeroperatoren durch Masse, Diplomarbeit, Bielefeld (1983)

    Google Scholar 

  15. J. Brasche, Perturbations of Schrödinger Hamiltonians by measures-self-adjointness and lower semiboundedness, ZiF-Preprint (1984) (to appear in J. Math. Phys.)

    Google Scholar 

  16. W. Karwowski, J. Marion, On the closability of some positive definite symmetric differential forms on C ∞0 (Ω), CNRS, Marseille, Preprint (1984)

    Google Scholar 

  17. S. Albeverio, R. Høegh-Krohn, L. Streit, Regularization of processes and Hamiltonians, J. Math. Phys. 21, 1636–1642 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Albeverio, S. Kusuoka, L. Streit, Convergence of Dirichlet forms and associated Schrödinger operators, BiBoS-Preprint (1984)

    Google Scholar 

  19. M. Fukushima, On a stochastic calculus related to Dirichlet forms and distorted Brownian motions, Phys. Repts. 77, 255–262 (1981)

    Article  MathSciNet  Google Scholar 

  20. I. G. Hooton, Dirichlet forms associated with hyper-contractive semigroups, Trans. Am. Math. Soc. 253, 237–256 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. N. Wielens, Eindeutigkeit von Dirichletformen und wesentliche Selbstadjungiertheit von Schrödingeroperatoren mit stark singulären Potentialen, Diplomarbeit, Bielefeld (1983)

    Google Scholar 

  22. N. Wielens, On the essential self-adjointness of generalized Schrödingeroperators, J. Funct. Anal., 1984

    Google Scholar 

  23. L. Streit, Energy forms: Schrödinger theory, processes, Phys. Repts. 77, 363–375 (1981)

    Article  MathSciNet  Google Scholar 

  24. S. Albeverio, R. Høegh-Krohn, Markov processes and Markov fields in quantum theory, group theory, hydrodynamics and C-algebras, pp. 497–540 in “Stochastic Integrals”, Ed. D. Williams, Lect. Notes Maths. 851, Springer, Berlin (1980)

    Chapter  Google Scholar 

  25. S. Albeverio, R. Høegh-Krohn, Some remarks on Dirichlet forms and their applications to quantum mechanics and statistical mechanics, pp. 120–132 in Functional Analysis in Markov-Processes, Ed. M. Fukushima, Lect. Notes Maths. 923, Springer, Berlin (1982)

    Chapter  Google Scholar 

  26. S. Albeverio, R. Høegh-Krohn, Diffusion fields, quantum fields and fields with values in groups in Adv. probability, Stochastic Analysis, Ed. M. Pinsky, Dekker, New York (1984)

    Google Scholar 

  27. R. Carmona, Processus de diffusion gouvernés par la forme de Dirichlet de l’opérateur de Schroedinger, Sém. Prob. XIII, Lect. Notes Maths. 721, Springer, Berlin (1979)

    Google Scholar 

  28. S. Albeverio, F. Gesztesy, W. Karwowski, L. Streit, On the connection between Schrödinger- and Dirichlet forms, BiBoS-Preprint (1984)

    Google Scholar 

  29. M. Nagasawa, Segregation of a population in an environment, J. Math. Biology 9, 213–235 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Albeverio, M. Fukushima, W. Karwowski, L. Streit, Capacity and quantum mechanical tunneling, Comm. Math. Phys. 80, 301–342 (1981)

    Article  Google Scholar 

  31. S. Albeverio, Ph. Blanchard, F. Gesztesy, L. Streit, Quantum mechanical low energy scattering in terms of diffusion processes, ZiF-Preprint 1984, to appear in Proc. II Enc. Math. Phys., France-Germany, Lect. Notes Maths., Springer (1984)

    Google Scholar 

  32. S. Albeverio, W. Karwowski, L. Streit, Some remarks on capacity, Green functions and Schrödinger operators, to appear in Ref. 11

    Google Scholar 

  33. M. Fukushima, Markov processes and functional analysis, pp. 187–202 in Proc. Int. Math. Conf., L.H.Y. Chen et al, Edts., North Holland (1982)

    Chapter  Google Scholar 

  34. M. Fukushima, A note on irreducibility and ergodicity of symmetric Markov processes, pp. 200–208 in “Stochastic processes in quantum hteory and statistical physics”, Edt. S. Albeverio et al, Lect. Notes Phys. 173, Springer, Berlin (1982)

    Chapter  Google Scholar 

  35. E. Carlen, Conservative diffusions and constructive approach to Nelson stochastic mechanics, Princeton Univ. Ph D Thesis (1984)

    Google Scholar 

  36. W.A. Cheng, Sur la méchanique stochastique, RMA Strasbourg (1984)

    Google Scholar 

  37. M. Fukushima, On distorted Brownian motions, pp. 255–262 Phys. Repts. 77, (1981)

    Article  MathSciNet  Google Scholar 

  38. M. Fukushima, On absolute continuity of multidimensional symmetrizable diffusions, pp. 146–176 in Functional Analysis in Markov-Processes, Ed. M. Fukushima, Lect. Notes Maths. 923, Springer, Berlin (1982)

    Chapter  Google Scholar 

  39. H. Nagai, Impulsive control of symmetric Markov processes and quasi-variational inequalities, Osaka J. Math. 20, 863–879 (1983)

    MathSciNet  MATH  Google Scholar 

  40. H. Nagai, Stopping problems of symmetric Markov processes and nonlinear variational inequalities, Preprint 1984

    Google Scholar 

  41. M. Fukushima, A generalized stochastic calculus in homo-geneization, pp.42–51 in “Quantum fields — Algebras, Processes”; Proc. Symp. Bielefeld, 1978, Ed. L. Streit, Süringer. Wien (1980)

    Google Scholar 

  42. H. Föllmer, Dirichlet processes, pp. 476–478 in “Stochastic Integrals”, Ed. D. Williams, Lect. Notes Maths. 851, Springer, Berlin (1980)

    Chapter  Google Scholar 

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

    MathSciNet  Google Scholar 

  44. M. Fukushima, M. Takeda, A transformation of a symmetric Markov processes and the Donsker-Varadhan theory, Osaka J. Math. 21, 311.326 (1984)

    MathSciNet  Google Scholar 

  45. M. Fukushima, M. Okada, On conformai martingales diffusions and pluripolar sets, J. Funct. Anal. 55, 377–388 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Albeverio, Ph. Blanchard, R. Høegh-Krohn, A stochastic model for the orbits of planets and satellites — The Titius-Bode law, Exp. Math. 4, 365–373 (1983)

    Google Scholar 

  47. S. Albeverio, Ph. Blanchard, R. Høegh-Krohn, Diffusions sur une variété Riemannienne: barrières infanchissables et applications, Coll. L. Schwartz, Astérisque 1984

    Google Scholar 

  48. S. Albeverio, Ph. Blanchard, Ph. Combe, R. Høegh-Krohn, R. Rodriguez, M. Sirugue, M. Sirugue-Collin, Magnetic bottles in a dirty environment. A stochastic model for radiation belts, ZiF-Preprint 1984

    Google Scholar 

  49. S. Albeverio, R. Høegh-Krohn, The structure of diffusion processes, Marseille, Preprint

    Google Scholar 

  50. S. Albeverio, R. Høegh-Krohn, Dirichlet forms and diffusion processes on rigged Hilbert spaces, Zeitschr. für Wahrscheinlichkeitstheorie verw. Gebiete 40, 1–57 (1977)

    Article  MATH  Google Scholar 

  51. S. Albeverio, R. Høegh-Krohn, Hunt processes and analytic potentials theory on rigged Hubert spaces, Ann. Inst. H. Poincaré B 13, 269–291 (1977)

    MATH  Google Scholar 

  52. S. Albeverio, R. Høegh-Krohn, Canonical quantum fields in two space-time dimensions, Preprint

    Google Scholar 

  53. S. Albeverio, R. Høegh-Krohn, Topics in infinite dimensional analysis, pp. 278–302 in Lect. Notes Phys. 80, Springer (1978) (Ed. G.F. Dell’Antonio et al)

    Google Scholar 

  54. Ph. Paclet, Espaces de Dirichlet et capacités fonctionnelles sur des triplets de Hilbert-Schmidt, Sém. Paul Krée,1977–1978 No. 5, 1–36

    Google Scholar 

  55. Ph. Paclet, Espaces de Dirichlet endimension infinie, CR. Ac.-Sciences Paris Ser. A 288, 981–993 (1979)

    MathSciNet  Google Scholar 

  56. M. Takeda, On the uniqueness of Markovian extension of diffusion operators on infinite dimensional spaces, ZiF-Preprint, 1984

    Google Scholar 

  57. S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lind-strøm-Non standard methods in stochastic analysis and mathematical physics, book in preparation (Acad. Press, (1985)

    Google Scholar 

  58. B. Simon, The P(φ)2 Euclidean (Quantum) field theory, Princeton Univ. (1974)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  61. B. Zegarlinski, Uniqueness and the global Markov property for Euclidean fields: the case of general exponential interactions, Comm. Math. Phys., to appear

    Google Scholar 

  62. S. Albeverio, R. Høegh-Krohn, Uniqueness and global Markov property for Euclidean fields and Lattice systems, pp. 303–329 in Quantum fields, Algebra Processes, Ed. L. Streit, Springer Verlag, Wien (1980)

    Google Scholar 

  63. S. Albeverio, R. Høegh-Krohn, Stochastic methods in quantum field theory and hydrodynamics, Phys. Repts. 77, 193–214 (1981)

    Article  Google Scholar 

  64. S. Albeverio, R. Høegh-Krohn, Local and global Markoff fields, Repts. Math. Phys. H, 225–248 (1984)

    Google Scholar 

  65. M. Röckner, Generalized Markov fields and axiomatic potential theory, Math. Ann. 264, 153–177 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  66. M. Röckner, Self-adjoint harmonic spaces and Dirichlet forms, Hiroshima Math. J., to appear

    Google Scholar 

  67. M. Röckner, A Dirichlet problem for distributions and the construction of specifications for Gaussian generalized random fields, Mem. Am. Math. Soc., to appear

    Google Scholar 

  68. M. Röckner, Generalized Markov fields and Dirichlet forms, Acta Applic. Math., to appear

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  70. T. Kolsrud, in preparation

    Google Scholar 

  71. S. Albeverio, R. Høegh-Krohn, D. Testard, Factoriality of representations of the group of paths of SU(n), J. Funct. Anal. 57, 49–55 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  72. J. Marion, D. Testard, Energy representations of gauge groups coming from foliations, Luminy, Preprint (1984)

    Google Scholar 

  73. S. Albeverio, R. Høegh-Krohn, Diffusion, quantum fields and groups of mappings, pp. 133–145 in Functional Analysis in Markov-Processes, Ed. M. Fukushima, Lect. Notes Maths. 923, Springer, Berlin (1982)

    Chapter  Google Scholar 

  74. L. Gross, Logarithmic Sobolov inequalities, Amer. J. Math. 97, 1061–1083 (1975)

    Article  Google Scholar 

  75. B. Lascar, Opérateurs pseudo-différentiels en dimension infinie. Etude de l’hypoellipticité et de la résolubilité dans des classes de fontions hölderiennes et de distribution pour des opérateurs pseudo-différentiels elliptiques, J. Anal. Math. 28, 39–104 (1978)

    Article  MathSciNet  Google Scholar 

  76. S. Albeverio, R. Høegh-Krohn, H. Holden, in preparation

    Google Scholar 

  77. S. Albeverio, R. Høegh-Krohn, H. Holden, Markov cosur-faces and gauge fields, Acta Phys. Austr. Suppl. 26, 211–231 (1984)

    Google Scholar 

  78. (Stochastic Methods and Computer Techniques in Quantum Dynmaics, Ed. H. Mitter, L. Pittner)

    Google Scholar 

  79. E. Seiler, Gauge theories as a problem of construction quantum field theory and statistical mechanics, Lect. Notes Phys. 159, Springer, Berlin (1982)

    Google Scholar 

  80. A. Kaufmann, in preparation

    Google Scholar 

  81. J. Brasche, BiBoS-Preprint, in preparation

    Google Scholar 

Download references

Authors
  1. S. Albeverio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Høegh-Krohn
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Holden
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, Germany

    L. Arnold  & P. Kotelenez  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1985 D. Reidel Publishing Company

About this chapter

Cite this chapter

Albeverio, S., Høegh-Krohn, R., Holden, H. (1985). Markov Processes on Infinite Dimensional Spaces, Markov Fields and Markov Cosurfaces. In: Arnold, L., Kotelenez, P. (eds) Stochastic Space—Time Models and Limit Theorems. Mathematics and Its Applications, vol 19. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5390-1_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-5390-1_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8879-4

  • Online ISBN: 978-94-009-5390-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation