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Probability Theory and Related Fields

, Volume 118, Issue 1, pp 37–48 | Cite as

Right inverses of Lévy processes and stationary stopped local times

Article

Abstract

If X is a symmetric Lévy process on the line, then there exists a non-decreasing, càdlàg process H such that X(H(x)) = x for all x≥ 0 if and only if X is recurrent and has a non-trivial Gaussian component. The minimal such H is a subordinator K. The law of K is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of X. When X is Brownian motion, K is just the usual ladder times process and this result extends the classical result of Lévy that the maximum process has the same law as the local time at 0. Write G t for last point in the range of K prior to t. In a parallel with classical fluctuation theory, the process Z := (X t X Gt ) t ≥0 is Markov with local time at 0 given by (X Gt ) t ≥0. The transition kernel and excursion measure of Z are identified. A similar programme is outlined for Lévy processes on the circle. This leads to the construction of a stopping time such that the stopped local times constitute a stationary process indexed by the circle.

Key words and phrases

Lévy process Local time Subordinator Fluctuation theory Reflected process 

Mathematics Subject Classification (2000)

Primary: 60J30, 60J55 Secondary: 60G10, 60J25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  1. 1.Department of Statistics #3860University of California at BerkeleyBerkeleyUSA

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