Necessary and Sufficient Conditions for the Continuity of Local Time of Levy Processes

Abstract

Let $u_1(x)$ be the 1-potential kernel density for a Levy process, let $\phi^2(x) = 2u_1(0) - u_1(x) - u_1(-x)$, let $\bar{\phi}$ be the monotone rearrangement of $\phi$ and let $I(\bar{\phi}) = \int_{0+} \phi(u)u^{-1}(\log(1/u))^{-1/2} du$. Barlow and Hawkes proved that if $I(\bar{\phi}) < \infty$, then the local time has a jointly continuous version. In this paper it is shown that if $I(\bar{\phi}) < \infty$, then the local time is not continuous.