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

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.