Right inverses of Lévy processes and stationary stopped local times (original) (raw)

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.