Right inverses of L\'evy processes and stationary stopped local times

August, 1999
Report Number: 
Steven N. Evans
Electronic Journal of Probability</em>, Vol. 5 (2000) Paper no. 7, pages 1-17

If $X$ is a 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 $x \ge 0$ 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 $Z := (X_t - X_{G_t})_{t \ge 0}$ is Markov with local time at $0$ given by $(X_{G_t})_{t \ge 0}$. The transition kernel and excursion measure of $Z$ are identified. A similar programme is carried out 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 indexed by the circle.

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