## The Annals of Probability

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

M. T. Barlow

#### 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.

#### Article information

Source
Ann. Probab., Volume 16, Number 4 (1988), 1389-1427.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991576

Digital Object Identifier
doi:10.1214/aop/1176991576

Mathematical Reviews number (MathSciNet)
MR958195

Zentralblatt MATH identifier
0666.60072

JSTOR