Local time of a diffusion in a stable Lévy environment

Abstract

Consider a 1-D diffusion in a stable Lévy environment. In this article, we prove that the normalized local time process recentred at the bottom of the standard valley with height log t, , converges in law to a functional of two independent Lévy processes, which are conditioned to stay positive. In the proof of the main result, we derive that the law of the standard valley is close to a two-sided Lévy process conditioned to stay positive. Moreover, we compute the limit law of the supremum of the normalized local time. In the case of a Brownian environment, similar result to the ones proved here have been obtained by Andreoletti and Diel.

Keywords:

• diffusion process
• local time
• Lévy process
• random environment

2000 Mathematics Subject Classification::

• 60G51
• 60J55
• 60J60

Acknowledgements

We are grateful to Romain Abraham and Pierre Andreoletti for helpful discussions. We also want to thank an anonymous referee for careful reading and many suggestions.

Notes

1. Email: [email protected]

Additional information

Notes on contributors

Roland Diel

1