On the Hilbert transform of the local times of a Lévy process (Q1897524)

Let \(\{X_t\}_{t \geq 0}\) be a real-valued Lévy process allowing for local times \(L^x_t\), denote by \(\tau (t)\) the inverse of \(L^0_t\), and by \(H_t = \pi^{-1} \int^\infty_0 (L^x_t - L_t^{-x}) x^{-1} dx\). The author gives explicit formulae for the Fourier-Laplace transforms \(E^0 (\exp \{-q \tau (t) + i \lambda H_{\tau (t)} \})\) and \(E^0 (\exp i (\lambda H_T + \xi X_T))\) - - here \(T\) is an independent exponential time -- essentially in terms of the characteristic exponent \(\psi\) of \(\{X_t\}_{t \geq 0}\). The proofs rely on Fourier analysis and the Feynman-Kac formula. The paper extends previous results by \textit{P. Biane} and \textit{M. Yor} [Bull. Sci. Math., II. Sér. 111, 23-101 (1987; Zbl 0619.60072)] and \textit{P. J. Fitzsimmons} and \textit{R. K. Getoor} [Ann. Probab. 20, No. 3, 1484-1497 (1992; Zbl 0767.60071)].