Uniform control of local times of spectrally positive stable processes

TL;DR

This paper provides uniform approximation and moment control results for local times of spectrally positive stable processes, with applications to branching processes and diffusion in complex spaces.

Contribution

It introduces new uniform approximation and moment control results for local times of spectrally positive stable processes, including a novel notion of restricted Lévy processes.

Findings

Established uniform approximation of local times in space and time.

Derived moment bounds on the Hölder constants of local times.

Applied results to construct path-continuous branching processes.

Abstract

We establish two results about local times of spectrally positive stable processes. The first is a general approximation result, uniform in space and on compact time intervals, in a model where each jump of the stable process may be marked by a random path. The second gives moment control on the H\"older constant of the local times, uniformly across a compact spatial interval and in certain random time intervals. For the latter, we introduce the notion of a L\'evy process restricted to a compact interval, which is a variation of Lambert's L\'evy process confined in a finite interval and of Pistorius' doubly reflected process. We use the results of this paper to exhibit a class of path-continuous branching processes of Crump-Mode-Jagers type with continuum genealogical structure. A further motivation for this study lies in the construction of diffusion processes in spaces of interval partitions and R-trees, which we explore in forthcoming articles.