Transition densities of one-dimensional Levy processes

TL;DR

This paper investigates the existence of transition densities for one-dimensional Lévy processes, extending previous results to include processes with Lévy symbols exhibiting logarithmic growth at infinity, including complex nested logarithmic forms.

Contribution

It broadens the class of Lévy processes known to have transition densities by analyzing symbols with logarithmic behavior, including nested logarithms, which were not covered before.

Findings

Established existence of transition densities for Lévy processes with logarithmic Lévy symbols.

Extended previous results to include complex nested logarithmic symbols.

Proved that certain nested logarithmic symbols correspond to Lévy processes with densities.

Abstract

In this paper, we study the existence of the transition densities of one-dimensional L\'evy processes. Compared with past results, our results contain the L\'evy processes whose L\'evy symbols have logarithm behavior at infinity. Our results contain the L\'evy symbol induced by the following Laplace exponent $\psi(\xi) := (\ln(1 + \ln(1 + \ln(\cdots \ln(1 + |\xi|)))))^\epsilon$ ($n$ times), $0 < \epsilon < 1$, $2 \le n$. We also show that $\psi(\xi)$ is a L\'evy symbol with transition density.