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

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

Report Number
565
Authors
Steven N. Evans
Citation
Electronic Journal of Probability</em>, Vol. 5 (2000) Paper no. 7, pages 1-17
Abstract

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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