Several long-time limit theorems of one-dimensional Lévy 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 Lévy process is recurrent and of finite variance.
The research of Shosei Takeda and Kouji Yano was supported by JSPS Open Partnership Joint Research Projects grant no. JPJSBP120209921. This research was supported by RIMS and by ISM. The research of Kouji Yano was supported by JSPS KAKENHI grant no.’s JP19H01791 and JP19K21834.
The authors would like to thank Hiroshi Tsukada for his helpful comments.
"Local time penalizations with various clocks for Lévy processes." Electron. J. Probab. 28 1 - 35, 2023. https://doi.org/10.1214/23-EJP903