Complements on the Hilbert transform and the fractional derivative of Brownian local times. (English) Zbl 0725.60084

Bertoin, Jean

J. Math. Kyoto Univ. 30, No. 4, 651-670 (1990).

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

Keywords:

Brownian motion; local times; Hilbert transform; fractional derivative; limit theorems for occupation times; analogues of Itô and Tanaka formulas; Ray-Knight type theorem; Bessel processes; excursion theory