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Transformation de Fourier et temps d'occupation browniens
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  • Published: June 1991

Transformation de Fourier et temps d'occupation browniens

  • C. Donati-Martin1 

Probability Theory and Related Fields volume 88, pages 137–166 (1991)Cite this article

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Summary

In this paper, we study oscillatory stochastic integrals of the form\(\Gamma (\lambda ) = \int\limits_0^\infty {exp(i \lambda B_s } )g(s)d s\) where λ is a non zero parameter andg a square integrable function. We study integrability properties of Γ(λ) and its behavior as a function of λ, using stochastic calculus techniques: martingale theory, representation of Itô for a random variable of the Wiener space, lemma of Garsia-Rodemich-Rumsey .... We also obtain limit theorems in law related to the variables Γ(λ) based upon an asymptotic version of a theorem of Knight on orthogonal continuous martingales.

We consider the random measure, image by the Brownian motion of the unbounded measure 1[0,∞] (s)g(s) ds; we prove the existence and the continuity of an occupation time density.

Finally, under a stronger integrability condition ong, we show the existence of a density for the law of Γ(λ), using Malliavin's calculus.

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Références

  • [B] Barlow, M.T.: Continuity of local times for Levy processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 23–35 (1985)

    Google Scholar 

  • [B-H] Bouleau, N., Hirsch, F.: Propriétés d'absolue continuité dans les espaces de Dirichlet et applications aux équations differentielles stochastiques. Séminaire de Probabilités XX (Lect. Notes Math., vol. 1204). Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  • [D] Donati-Martin, C.: Le problème de Buffon-Synge pour une corde. Adv. Appl. Probab.22, 375–395 (1990)

    Google Scholar 

  • [D1] Donati-Martin, C.: Deux études sur le mouvement brownien. Thèse d'Université Paris VI (février 1989)

  • [Fe] Feller, W.: An introduction to probability theory and its applications. New York: Wiley 1981

    Google Scholar 

  • [Fr] Freedman, D.: On tail probabilities for martingales. Ann. Probab.3, 100–118 (1975)

    Google Scholar 

  • [G] Garsia, A.: Continuity properties of multi-dimensional Gaussian processes. 6th Berkeley Symposium on Math. Probab., vol. 2, pp. 369–376, Berkeley (1970)

    Google Scholar 

  • [G-R-R] Garsia, A., Rodemich, E., Rumsey, H.: A real lemma and the continuity of paths of some gaussian processes. Indiana Univ. Math. J.20, 565–578 (1970)

    Google Scholar 

  • [J] Jeulin, T. Semi-martingales et grossissement d'une filtration (Lect. Notes Math., vol. 833), Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  • [K-K] Kasahara, Y., Kotani, S.: On limit processes for a class of additive functionals of recurrent diffusion processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 133–153 (1979)

    Google Scholar 

  • [Ki] Kingman, J.F.C.: The thrown string. J.R. Stat. Soc. Ser. B44, 109–138 (1982)

    Google Scholar 

  • [Kn] Knight, F.B.: A reduction of continuous square integrable Martingales to Brownien motion (Lect. Notes Math., vol. 190). Berlin Heidelberg New York: Springer 1971

    Google Scholar 

  • [L-Y] Le Gall, J.F., Yor, M.: Etude asymptotique de certains mouvements browniens complexes avec drift. Probab. Th. Rel. Fields71, 183–229 (1986)

    Google Scholar 

  • [N-P] Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Th. Rel. Fields78, 535–581 (1988)

    Google Scholar 

  • [P-S-V] Papanicolaou, G.C., Stroock, D.W. Varadhan, S.R.S.: Martingale approach to some limit theorems. Duke Univ. Math. Ser. III, Statistical mechanics and dynamical systems (1977)

  • [P-Y] Pitman, J., Yor, M.: Asymptotic laws of planar Brownian motion. Ann. Probab.14, 733–779 (1986)

    Google Scholar 

  • [S-V] Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  • [Y] Yor, M.: Le drap brownien comme limite en loi de temps locaux linéaires. Séminaire de probabilités XVII (Lect. Notes Math., vol. 986). Berlin Heidelberg New York: Springer 1983

    Google Scholar 

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  1. Université de Provence, URA 225, 3, place Victor Hugo, F-13331, Marseille Cedex 3, France

    C. Donati-Martin

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Donati-Martin, C. Transformation de Fourier et temps d'occupation browniens. Probab. Th. Rel. Fields 88, 137–166 (1991). https://doi.org/10.1007/BF01212557

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  • Received: 02 October 1989

  • Revised: 04 October 1990

  • Issue Date: June 1991

  • DOI: https://doi.org/10.1007/BF01212557

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