Right Inverses Of Lévy Processes And Stationary Stopped Local Times

. If X is a symmetric L'evy process on the line, then there exists a non--decreasing, c`adl`ag 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 multiple 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 L'evy 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 \Gamma XG t ) t0 is Markov with local time at 0 given by (XG t ) t0 . The transition kernel and excursion measure of Z are identified. A similar programme is outlined for L'evy processes on the circle. This leads to the construction of a stopping time such that the stopped local times constitute a stationary process ..