Transformation de Fourier et temps d'occupation browniens. (Fourier transformation and Brownian occupation time)
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 \textit{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.
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