A note on the existence of transition probability densities for L\'evy processes

TL;DR

This paper establishes conditions for the existence of smooth transition probability densities in Lévy processes, analyzes their asymptotic behavior, and proves a ratio-limit theorem, enhancing understanding of their probabilistic structure.

Contribution

It provides necessary and sufficient conditions for the existence of smooth densities in Lévy processes and characterizes their asymptotic behavior.

Findings

Conditions for existence of smooth densities derived

Asymptotic behavior of densities calculated as time approaches zero and infinity

A ratio-limit theorem for Lévy processes proved

Abstract

We prove several necessary and sufficient conditions for the existence of (smooth) transition probability densities for L\'evy processes and isotropic L\'evy processes. Under some mild conditions on the characteristic exponent we calculate the asymptotic behaviour of the transition density as $t\to 0$ and $t\to\infty$ and show a ratio-limit theorem.