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Complements on the Hilbert transform and the fractional derivative of Brownian local times. (English) Zbl 0725.60084
Let B be a real Brownian motion and $$(L^ a_ t:$$ $$a\in {\mathbb{R}}$$, $$t\geq 0)$$ its jointly continuous local times. Since $$a\mapsto L^ a_ t$$ is a.s. Hölder continuous of order 1/2-$$\epsilon$$ for all positive $$\epsilon$$, one can introduce its Hilbert transform $C(t)=\int^{\infty}_{0}(L^ a_ t-L_ t^{-a})a^{-1}da,$ and its fractional derivative of order $$\alpha\in]0,1/2[$$ $H(-1- \alpha,t)=\int^{\infty}_{0}(L^ a_ t-L^ 0_ t)a^{-1- \alpha}da.$ T. Yamada [ibid. 26, 309-322 (1986; Zbl 0618.60080)] proved that these processes appear in limit theorems for occupation times of Brownian motion. They have an unbounded variation, but a bounded p- variation for some suitable $$p\in]1,2[$$. The purpose of this paper is to study C and H by using the extension of the classical stochastic calculus for processes with bounded p-variation, which was developed by the author [Ann. Probab. 17, No.4, 1521-1535 (1989; Zbl 0687.60054)]. One gets analogues of Itô and Tanaka formulas. The main result is a Ray-Knight type theorem which describes the occupation densities of H taken at its first hitting time of a given level, in terms of squares of Bessel processes. The author [Probab. Theory Relat. Fields 84, No.2, 231-250 (1990; Zbl 0665.60073)] has recently explained and extended this result by excursion theory.
Reviewer: J.Bertoin (Paris)

##### MSC:
 60J55 Local time and additive functionals
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