Cauchy’s principal value of local times of Lévy processes with no negative jumps via continuous branching processes

Let X be a recurrent Lévy process with no negative jumps and n the measure of its excursions away from 0. Using Lamperti’s connection [13] that links X to a continuous state branching process, we determine the joint distribution under n of the variables C T = ∫ T 0 1{Xs>0}X −1 s ds and C− T = ∫ T 0 1{Xs<0}|Xs|ds, where T denotes the duration of the excursion. This provides a new insight on an identity of Fitzsimmons and Getoor [9] on the Hilbert transform of the local times of X . Further results in the same vein are also discussed.