Local time penalizations with various clocks for Levy processes

Several long-time limit theorems of one-dimensional Levy processes weighted and normalized by functions of the local time are studied. The long-time limits are taken via certain families of random times, called clocks: exponential clock, hitting time clock, two-point hitting time clock and inverse local time clock. The limit measure can be characterized via a certain martingale expressed by an invariant function for the process killed upon hitting zero. The limit processes may differ according to the choice of the clocks when the original Levy process is recurrent and of finite variance.

Automated summary

This study examines several long-time limit theorems for one-dimensional Levy processes that are weighted and normalized by functions of the local time. These long-time limits are obtained through different families of random times, namely exponential clock, hitting time clock, two-point hitting time clock, and inverse local time clock. The characterization of the limit measure is achieved by a martingale expressed by an invariant function for the process killed when hitting zero. The choice of clocks may lead to different limit processes in the case of a recurrent and finite variance Levy process.