×

Transformation de Fourier et temps d’occupation browniens. (Fourier transformation and Brownian occupation time). (French) Zbl 0717.60092

We study oscillatory stochastic integrals of the form \[ \Gamma (\lambda)=\int^{\infty}_{0}\exp (i\lambda B_ s)g(s)ds, \] where \(\lambda\) is a nonzero parameter and g a square integrable function. We study integrability properties of \(\Gamma\) (\(\lambda\)) and its behavior as a function of \(\lambda\), 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 \(\Gamma\) (\(\lambda\)) based upon an asymptotic version of a theorem of F. B. Knight [A reduction of continuous square integrable martingales to Brownian motion, in: H. Dinges (ed.), Martingales. A report on a meeting at Oberwolfach, May 17- 23, 1970 (1971; Zbl 0226.60070)] on orthogonal continuous martingales.
We consider the random measure, image by the Brownian motion of the unbounded measure \(1_{[0,\infty]}(s)g(s)ds\); we prove the existence and the continuity of an occupation time density. Finally, under a stronger integrability condition on g, we show the existence of a density for the law of \(\Gamma\) (\(\lambda\)), using Malliavin’s calculus.
Reviewer: C.Donati-Martin

MSC:

60J65 Brownian motion
60J55 Local time and additive functionals

Citations:

Zbl 0226.60070
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] [B] Barlow, M.T.: Continuity of local times for Levy processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 23-35 (1985) · Zbl 0561.60076
[2] [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
[3] [D] Donati-Martin, C.: Le problème de Buffon-Synge pour une corde. Adv. Appl. Probab.22, 375-395 (1990) · Zbl 0701.60009
[4] [D1] Donati-Martin, C.: Deux études sur le mouvement brownien. Thèse d’Université Paris VI (février 1989)
[5] [Fe] Feller, W.: An introduction to probability theory and its applications. New York: Wiley 1981 · Zbl 0077.12201
[6] [Fr] Freedman, D.: On tail probabilities for martingales. Ann. Probab.3, 100-118 (1975) · Zbl 0313.60037
[7] [G] Garsia, A.: Continuity properties of multi-dimensional Gaussian processes. 6th Berkeley Symposium on Math. Probab., vol. 2, pp. 369-376, Berkeley (1970)
[8] [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) · Zbl 0252.60020
[9] [J] Jeulin, T. Semi-martingales et grossissement d’une filtration (Lect. Notes Math., vol. 833), Berlin Heidelberg New York: Springer 1980 · Zbl 0444.60002
[10] [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) · Zbl 0435.60080
[11] [Ki] Kingman, J.F.C.: The thrown string. J.R. Stat. Soc. Ser. B44, 109-138 (1982) · Zbl 0545.60019
[12] [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
[13] [L-Y] Le Gall, J.F., Yor, M.: Etude asymptotique de certains mouvements browniens complexes avec drift. Probab. Th. Rel. Fields71, 183-229 (1986) · Zbl 0579.60077
[14] [N-P] Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Th. Rel. Fields78, 535-581 (1988) · Zbl 0629.60061
[15] [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) · Zbl 0387.60067
[16] [P-Y] Pitman, J., Yor, M.: Asymptotic laws of planar Brownian motion. Ann. Probab.14, 733-779 (1986) · Zbl 0607.60070
[17] [S-V] Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979 · Zbl 0426.60069
[18] [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.