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

August, 1999
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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