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The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in Rd
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  • Published: March 1994

The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in Rd

  • Peter Imkeller1 &
  • Ferene Weisz2 

Probability Theory and Related Fields volume 98, pages 47–75 (1994)Cite this article

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  • 18 Citations

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Summary

LetL(x, T),x∈R d,T∈R N+ , be the local time of theN-parameter Wiener processW taking values inR d. Even in the distribution valued casedd≧2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|→0 and/orT∞ and of related occupation integrals\(X_T (f) = \int\limits_{[0,T]} f (W_S )\) asT→∞. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.X T (f), and of the second order, i.e. normalized convergence laws forL(x, T)−E(L(x, T)) resp.X T (f)−E(X T (f)).

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References

  1. Berman, S.M.: Gaussian processes with stationary increments: local times and sample function properties. Ann. Math. Stat.41, 1260–1272 (1970)

    Google Scholar 

  2. Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. Berlin: de Gruyter 1991

    Google Scholar 

  3. Cairoli, R., Walsh, J.B.: Stochastic integrals in the plane. Acta Math.134, 111–183 (1975)

    Google Scholar 

  4. Dozzi, M.: On the local time of the multi-parameter Wiener process and the asymptotic behaviour of an associated integral. Stochastics25, 155–169 (1988)

    Google Scholar 

  5. Dozzi, M.: Stochastic processes with multidimensional parameter. Habilitationsschrift, University of Bern (1986)

  6. Ehm, W.: Sample function properties of multi-parameter stable processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 195–228 (1981)

    Google Scholar 

  7. Imkeller, P.: Stochastic analysis and local times for (N, d)-Wiener process. Ann. Inst. Henri Poincaré20, 75–101 (1984)

    Google Scholar 

  8. Imkeller, P., Perez-Abreu, V., Vives, J.: Chaos expansions of double intersection local time of Brownian motion inR d and renormalization. (Preprint 1992)

  9. Imkeller, P., Schmidt, W.: Stochastic integration for some rough non-adapted processes. Math. Nachr. (to appear)

  10. Kallianpur, G., Robbins, H.: Ergodic property of the Brownian motion process. Proc. Natl. Acad. Sci. USA39, 525–533 (1953)

    Google Scholar 

  11. Le Gall, J.F.: Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan. In: Azéma, J., Yor, M. (eds) Sém, de Prob. XIX, 1983/84. (Lect. Notes Math., vol. 1123, pp. 314–331) Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  12. Nualart, D., Vives, J: Chaos expansion and local times. University of Barcelona (Preprint 1992)

  13. Pitman, J., Yor, M.: Aymptotic laws of planar Brownian motion, Ann. Probab.11, 733–779 (1986)

    Google Scholar 

  14. Varadhan, S.R.S.: Appendix to “Euclidean quantum field theory” by K. Szymanzik. In: Jost, R. (ed.) Local quantum theory. New York: Academic Press 1969

    Google Scholar 

  15. Walsh, J.B.: The local time of the Brownian sheet. Astérisque52–53, 47–61 (1978)

    Google Scholar 

  16. Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematisches Institut der LMU München, Theresienstrasse 39, D-80333, München, Germany

    Peter Imkeller

  2. Department of Numerical Analysis, Eötvös L. University, Bogdánfy u. 10/b, H-1117, Budapest, Hungary

    Ferene Weisz

Authors
  1. Peter Imkeller
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  2. Ferene Weisz
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Additional information

This research was made during a stay at the LMU in München supported by DAAD

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Imkeller, P., Weisz, F. The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in Rd . Probab. Th. Rel. Fields 98, 47–75 (1994). https://doi.org/10.1007/BF01311348

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  • Received: 23 February 1993

  • Revised: 22 June 1993

  • Issue Date: March 1994

  • DOI: https://doi.org/10.1007/BF01311348

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Mathematics Subject Classification (1990)

  • 60G60
  • 60J55
  • 60G15
  • 60H05
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