Transition densities of one-dimensional Levy processes

Abstract

In this paper, we study the existence of the transition densities of one-dimensional Lévy processes. Compared with past results, our results contain the Lévy processes whose Lévy symbols have logarithm behavior at infinity. Our results contain the Lévy 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 Lévy symbol with transition density.