Local time penalizations with various clocks for L\'{e}vy processes

TL;DR

This paper investigates the long-time behavior of one-dimensional Lévy processes weighted by local time functions, using various types of random clocks to understand their limit distributions and the effects of different clock choices.

Contribution

It introduces a framework for analyzing long-time limits of Lévy processes with local time weights using diverse clocks, providing new characterizations of limit measures and processes.

Findings

Limit measures are characterized via martingales involving invariant functions.

Limit processes vary depending on the choice of clock for recurrent, finite variance Lévy processes.

Different clocks lead to different long-time limit behaviors of the processes.

Abstract

Several long-time limit theorems of one-dimensional L\'{e}vy 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 L\'{e}vy process is recurrent and of finite variance.