## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 30, Issue 1 (1986), 19-65.

### Two uniform intrinsic constructions for the local time of a class of Lévy processes

Martin T. Barlow, Edwin A. Perkins, and S. James Taylor

#### Abstract

We show that if $X$ is a Lévy process with a regularly varying exponent function and a local time, $L_{t}^{x}$, that satisfies a mild continuity condition, then for an appropriate function $\phi$, $$\phi-m\{s \leq t|X_{s} = x\} =L_{t}^{x}\quad \forall t \geq 0, \,x \in \mathbf{R}\,\,\,\sy{a.s.}$$ Here $\phi-m(E)$ denotes the Hausdorff $\phi$-measure of the set $E$. In particular if $X$ is a stable process of index $\alpha >1$, this solves a problem of Taylor and Wendel. We also prove that under essentially the same conditions, a construction of $L_{t}^{0}$ due to Kingman, in fact holds uniformly over all levels.

#### Article information

**Source**

Illinois J. Math., Volume 30, Issue 1 (1986), 19-65.

**Dates**

First available in Project Euclid: 20 October 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1256044751

**Digital Object Identifier**

doi:10.1215/ijm/1256044751

**Mathematical Reviews number (MathSciNet)**

MR822383

**Zentralblatt MATH identifier**

0571.60082

**Subjects**

Primary: 60J30

Secondary: 60G17: Sample path properties

#### Citation

Barlow, Martin T.; Perkins, Edwin A.; Taylor, S. James. Two uniform intrinsic constructions for the local time of a class of Lévy processes. Illinois J. Math. 30 (1986), no. 1, 19--65. doi:10.1215/ijm/1256044751. https://projecteuclid.org/euclid.ijm/1256044751