It is well known that a class of subordinators can be represented using the local time of Brownian motions. An extension of such a representation is given for a class of Lévy processes which are not necessarily of bounded variation. This class can be characterized by the complete monotonicity of the Lévy measures. The asymptotic behavior of such processes is also discussed and the results are applied to the generalized arc-sine law, an occupation time problem on the positive side for one-dimensional diffusion processes.
"Brownian representation of a class of Lévy processes and its application to occupation times of diffusion processes." Illinois J. Math. 50 (1-4) 515 - 539, 2006. https://doi.org/10.1215/ijm/1258059484