## The Annals of Applied Probability

### Uniform control of local times of spectrally positive stable 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 Hölder 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 Lévy process restricted to a compact interval, which is a variation of Lambert’s Lévy 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 $\mathbb{R}$-trees, which we explore in forthcoming articles. In that context, local times correspond to branch lengths.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 2592-2634.

Dates
Revised: August 2017
First available in Project Euclid: 9 August 2018

https://projecteuclid.org/euclid.aoap/1533780281

Digital Object Identifier
doi:10.1214/17-AAP1370

Mathematical Reviews number (MathSciNet)
MR3843837

Zentralblatt MATH identifier
06974759

#### Citation

Forman, Noah; Pal, Soumik; Rizzolo, Douglas; Winkel, Matthias. Uniform control of local times of spectrally positive stable processes. Ann. Appl. Probab. 28 (2018), no. 4, 2592--2634. doi:10.1214/17-AAP1370. https://projecteuclid.org/euclid.aoap/1533780281

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