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Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes
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  • Published: March 1995

Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes

  • F. Hirsch1 &
  • S. Song1 

Probability Theory and Related Fields volume 103, pages 25–43 (1995)Cite this article

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Summary

We introduce a new Skorohod topology for functions of several variables. Since ann-variable function may be viewed as a one-variable function with values in the set of (n−1)-variable functions, this topology is defined by induction from the classical Skorohod topology for one-variable functions. This allows us to define the notion of completen-parameter symmetric Markov processes: Such processes are, for any 1≤p≤n, rawp-parameter Markov processes (in the sense of our previous paper [17]) with values in the space of (n−p)-variable functions. We prove, for these processes and their Bochner subordinates, a maximal inequality which implies the continuity of additive functionals associated with finite energy measures. We finally present several important examples.

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References

  1. Bauer, J.: Multiparameter processes associated with Ornstein-Uhlenbeck semigroups. In: Gowrisankaran et al. (eds.) Classical and modern potential theory and applications. Dordrecht: Kluwer pp. 41–55, 1994

    Google Scholar 

  2. Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

    Google Scholar 

  3. Cairoli, R.: Produits de semi-groupes de transition et produits de processus. Publ. Inst. Stat. Univ. Paris (no 15) pp. 311–384, 1966

    Google Scholar 

  4. Dynkin, E.: Additive functionals of several time-reversible Markov processes. J. Funct. Anal.42, 64–101 (1981)

    Google Scholar 

  5. Dynkin, E.: Harmonic functions associated with several Markov processes. Adv. Appl. Math.2, 260–283 (1981)

    Google Scholar 

  6. Evans, N.: Potential theory for a family of several Markov processes. Ann. Inst. Henri Poincare23, 499–530 (1987)

    Google Scholar 

  7. Feyel, D.: Espaces de Banach adaptés, Quasi-topologie et balayage. In: Séminaire de theorie du potentiel Paris (no 3) pp. 499–530, 1987

  8. Feyel, D., de La Pradelle, A.: Sur les draps de dimension infinie (Preprint, 1994)

  9. Feyel, D., de La Pradelle, A.: Capacités gaussiennes. Ann. Inst. Fourier, Grenoble41, 49–76 (1991)

    Google Scholar 

  10. Fitzsimmons, P., Salisbury, T.: Capacity and energy for multiparameter Markov processes. Ann. Inst. Henri Poincaré25, 325–350 (1989)

    Google Scholar 

  11. Fukushima, M., Kaneko, H.: On (r, p)-capacities for general Markovian semi-groups. In: Infinite dimensional analysis and stochastic processes, pp. 41–47. Boston-London-Melbourne: Pitman (1985)

    Google Scholar 

  12. Gallardo, A.: Topologia estocastica para campos aleatorios. Tesis de doctor, Pontificia universidad catolica de Chile. (1991)

  13. He, S., Wang, J., Yan, J.: Semimartingale theory and stochastic calculus. Boca Raton: Science Press, CRC Press (1992)

    Google Scholar 

  14. Hirsch, F.: Représentation du processus d'Ornstein-Uhlenbeck àn-paramètres. In: Séminaire de Probabilités XXVII, (Lect. Notes Math., vol. 1557, pp. 302–303) Berlin: Springer 1993

    Google Scholar 

  15. Hirsch, F.: Potential theory related to some multiparameter processes. Potential Anal. 4: 245–267, 1995

    Google Scholar 

  16. Hirsch, F., Song, S.: Inequalities for Bochner's subordinates of two-parameter symmetric Markov processes Ann. I.H.P. (to appear)

  17. Hirsch, F., Song, S.: Markov properties of multiparameter processes and capacities. (1994) P.T.R.F. (to appear)

  18. Imkeller, P.: Two-parameter martingales and their quadratic variation (Lect. Notes Math., vol. 1308) Berlin: Springer 1988

    Google Scholar 

  19. Kasumi, T., Shigekawa, I.: Measures of finite (r, p)-energy and potentials on a separable metric space. In: Séminaire de Probabilités XXVI (Lect. Notes Math., vol. 1526, pp. 415–444) Berlin: Springer 1992

    Google Scholar 

  20. Maisonneuve, B.: Topologie du type de Skorohod. in Séminaire de Probabilités VI (Lect. Notes Math., vol. 258, pp. 113–117) Berlin: Springer 1983

    Google Scholar 

  21. Malliavin, P.: Implicit functions in finite corank on the Wiener space. In: Taniguchi Int. Symp. Stoch. Anal. Katata 1982, Kinokuniya, Tokyo, pp. 369–386, 1983

    Google Scholar 

  22. Mazziotto, G.: Two-parameter Hunt processes and a potential theory. The Ann. Probab.16, 600–619 (1988)

    Google Scholar 

  23. Mishura, Y.: On the convergence of random fields in the J-topology. Theoret. Probab. Math. Statist.17, 111–119 (1979)

    Google Scholar 

  24. Pagès, G.: Théorème limites fonctionnels pour les semi-martingales. Thèse du 3eme cycle de l'Université Paris VI. (1987)

  25. Prum, B.: Semi-martingales à indice dansR 2. Thèse de l'Université Paris-Sud Centre d'Orsay (1980)

  26. Ren, J.: Topologie p-fine sur l'espace de Wiener et théorème des fonctions implicites. Bull. Sci. Math., 2e série,114, 99–114 (1990)

    Google Scholar 

  27. Song, S.: Construction d'un processus à deux paramètres à partir d'un semi-groupe à un paramètre. In: Gowrisankaran et al. eds., Classical and modern potential theory and applications, pp. 419–451. Dordrecht: Kluwer Acad. Publ. 1994

    Google Scholar 

  28. Song, S.: Ihégalités relatives aux processus d'Ornstein-Uhlenbeck àn-paramètres et capacité gaussienneC n,2. In: Séminaire de Probabilités XXVII (Lect. Notes Math., vol. 1557, pp. 276–301) Berlin: Springer (1993)

    Google Scholar 

  29. Sugita, H.: Positive generalized Wiener functions and potential theory over abstract Wiener spaces. Osaka J. Math.25, 665–696 (1988)

    Google Scholar 

  30. Walsh, J.: An introduction to stochastic partial differential equations. In: Ecole d'été de probabilités de Saint-Flour XIV-1984 (Lect. Notes Math. vol 1180, pp. 266–437) Berlin: Springer 1986

    Google Scholar 

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Authors and Affiliations

  1. Equipe d'Analyse et Probabilités, Boulevard des Coquibus, F-91025, Evry Cedex, France

    F. Hirsch & S. Song

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  1. F. Hirsch
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  2. S. Song
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Hirsch, F., Song, S. Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes. Probab. Th. Rel. Fields 103, 25–43 (1995). https://doi.org/10.1007/BF01199030

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  • Received: 01 November 1994

  • Revised: 03 April 1995

  • Issue Date: March 1995

  • DOI: https://doi.org/10.1007/BF01199030

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Mathematics Subject Classification

  • 31C15
  • 60J25
  • 60J30
  • 60J45
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