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
Citation
Noah Forman. Soumik Pal. Douglas Rizzolo. Matthias Winkel. "Uniform control of local times of spectrally positive stable processes." Ann. Appl. Probab. 28 (4) 2592 - 2634, August 2018. https://doi.org/10.1214/17-AAP1370
Information