Spatial growth-fragmentations and excursions from hyperplanes

Abstract

In this paper, we are interested in the self-similar growth-fragmentation process that shows up when slicing half-space excursions of a $d$-dimensional Brownian motion from hyperplanes. Such a family of processes turns out to be a collection of spatial self-similar growth-fragmentation processes driven by an isotropic self-similar Markov process. The former can be seen as multitype growth-fragmentation processes, in the sense of Da Silva and Pardo (2024), where the set of types is $S^{d-2}$, the $(d-1)$–dimensional unit sphere. In order to characterise such family of processes, we study their spinal description similarly as in the monotype (Bertoin, 2017) and multitype (Da Silva and Pardo, 2024) settings. Finally, we extend our study to the case when the $d$-dimensional Brownian motion is replaced by an isotropic Markov process whose first $(d-1)$ coordinates are driven by an isotropic stable Lévy process and the remaining coordinate is an independent standard real-valued Brownian motion.