The asymptotic behavior of the renormalized zero resolvent of L\'evy processes under regular variation conditions

TL;DR

This paper investigates the asymptotic behavior near zero of the renormalized zero resolvent for one-dimensional Lévy processes, under regular variation assumptions on key process parameters, extending known stable case results.

Contribution

It provides new asymptotic formulas for the resolvent of Lévy processes under regular variation, generalizing explicit stable case formulas to broader classes.

Findings

Asymptotic behavior characterized at the origin

Extension of stable case explicit formulas

Conditions on Lévy measure and exponent identified

Abstract

As an analogue to the explicit formula in the stable case, the asymptotic behavior at the origin of the renormalized zero resolvent of one-dimensional L\'evy processes is studied under certain regular variation conditions on the L\'evy-Khinchin exponent and the L\'evy measure.