Precise asymptotic approximations for the kernels corresponding to the L\'evy processes

TL;DR

This paper develops precise asymptotic approximations for kernels of symmetric alpha-stable and general radial Lévy processes using complex analysis, enhancing understanding of their behavior.

Contribution

It introduces a novel complex analysis method to derive asymptotic approximations for Lévy process kernels, applicable to a broad class of processes.

Findings

Asymptotic formulas for symmetric alpha-stable kernels

Extensions to general radial Lévy processes

Improved accuracy in kernel approximations

Abstract

Using complex analysis techniques we obtain precise asymptotic approximations for the kernels corresponding to the symmetric $\alpha$-stable processes and their fractional derivatives. We apply our method to general L\'evy processes whose characteristic functions are radial and satisfy some regularity and size conditions.