Uniform control of local times of spectrally positive stable processes

Journal article

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 R-trees, which we explore in forthcoming articles. In that context, local times correspond to branch lengths.

Authors: Forman, N; Pal, S; Rizzolo, D; Winkel, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Mathematical Statistics
• Journal: Annals of Applied Probability
• Volume: 28
• Issue: 4
• Pages: 2592-2634
• Publication date: 2018-08-09
• Acceptance date: 2017-11-09
• DOI: 10.1214/17-AAP1370
• ISSN: 1050-5164