Compound kernel estimates for the transition probability density of a L\'evy process in $\rn$

TL;DR

This paper develops precise upper and lower estimates for the transition probability density of a Lévy process in multi-dimensional space, using complex analysis and asymptotic methods to analyze the inverse Fourier transform of its characteristic function.

Contribution

It introduces a novel approach combining complex analysis and asymptotic analysis to derive small-time estimates for Lévy process densities.

Findings

Established upper and lower bounds for the transition density

Applied complex analysis techniques to Lévy processes

Provided asymptotic descriptions of the inverse Fourier transform

Abstract

We construct in the small-time setting the upper and lower estimates for the transition probability density of a L\'evy process in $\rn$. Our approach relies on the complex analysis technique and the asymptotic analysis of the inverse Fourier transform of the characteristic function of the respective process.