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Potential Analysis

, Volume 4, Issue 3, pp 245–267 | Cite as

Potential theory related to some multiparameter processes

  • Francis Hirsch
Article

Abstract

In a first part, we present a potential theory constructed form a continuous kernel on a locally compact space. The notions of capacity, quasi-continuity, equilibrium measures and potentials are specially studied. In a second part, we particularize the framework, and, in the third part, we give probabilistic interpretations in this particular case. The process then involved is a sum of independent symmetric Levy processes in ℝ d , viewed as a multiparameter process. For instance, hitting probabilities for the process are estimated in terms of capacity.

Mathematics Subject Classifications (1991)

Primary 31C15 60J30 60J45 Secondary 60J55 

Key words

Potential theory capacity multiparameter process 

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References

  1. 1.
    J. Deny: ‘Théorie de la capacité dans les espaces fonctionnels', inSéminaire Brelot-Choquet-Deny (Théorie du Potentiel), 9ème année, 1964/65, no 1, 13 pages.Google Scholar
  2. 2.
    E. B. Dynkin: ‘Additive functionals of several time-reversible Markov processes’,J. Funct. Anal. 42 (1981), 64–101.Google Scholar
  3. 3.
    S. N. Evans: ‘Potential theory for a family of several Markov processes’,Ann. Inst. Henri Poincaré 23(3) (1987), 499–530.Google Scholar
  4. 4.
    D. Feyel: ‘Espaces de Banach adaptés. Quasi-topologie et balayage’, inSéminaire de théorie du potentiel Paris, No. 3, L.N. in Maths 681, Springer 1978, 81–102.Google Scholar
  5. 5.
    P. J. Fitzsimmons and T. S. Salisbury: ‘Capacity and energy for multiparameter Markov Processes’,Ann. Inst. Henri Poincaré 25(3) (1989), 325–350.Google Scholar
  6. 6.
    D. Geman and J. Horowitz: ‘Occupation densities’,Ann. of Proba. 8(1) (1980), 1–67.Google Scholar
  7. 7.
    F. Hirsch: ‘Capacité associée à une somme de processus indépendants à accroissements indépendants’,C.R. Acad. Sci. Paris 316,Série I (1993), 925–928.Google Scholar
  8. 8.
    F. Hirsch: ‘Représentation du processus d'Ornstein-Uhlenbeck àn paramètres’, inSéminaire de Probabilités XXVII, L.N. in Maths 1557, Springer-Verlag (1993), 302–303.Google Scholar
  9. 9.
    T. Kazumi and I. Shigekawa: ‘Measures of finite (r, p)-energy and potentials on a separable metric space,’ inSéminaire de Probabilités XXVI, L.N. in Maths 1526, Springer-Verlag (1992), 415–444.Google Scholar
  10. 10.
    Y. Le Jan: ‘Quasi-continuous functions and Hunt processes’,J. Math. Soc. Japan 35(1) (1983), 37–42.Google Scholar
  11. 11.
    G. Mokobodzki: ‘Capacités fonctionnelles’, inSéminaire Choquet (Initiation à l'Analyse), 6ème année, 1966/67, no 1, 6 pages.Google Scholar
  12. 12.
    S. Song: ‘Processus d'Ornstein-Uhlenbeck et ensemblesW 2,2-polaires’,Potential Analysis 2 (1983), 171–186.Google Scholar
  13. 13.
    S. Song: ‘Inégalités relatives aux processus d'Ornstein-Uhlenbeck àn-paramètres et capacité gaussiennec n, 2’, inSéminaire de Probabilités XXVII, L.N. in Maths 1557, Springer-Verlag (1993), 276–301.Google Scholar
  14. 14.
    S. Song: ‘Some results on potential theory forn-parameter Ornstein-Uhlenbeck process’, to appear.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Francis Hirsch
    • 1
  1. 1.Equipe d'Analyse et ProbabilitésUniversité d'Evry Val d'EssonneEvry cedexFrance

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