Uniform control of local times of spectrally positive stable processes

2016/09/21 by Forman, Noah, Pal, Soumik, Rizzolo, Douglas +1
#60G51 #60G52 #60J55 #60J80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1609.06707

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 R-trees, which we explore in forthcoming articles.

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