The extremal process of super-Brownian motion: a probabilistic approach via skeletons

TL;DR

This paper introduces a probabilistic approach using skeleton decomposition to analyze the extremal process of super-Brownian motion, providing new insights and results beyond previous PDE-based methods.

Contribution

It develops a novel skeleton-based method to study the extremal process of super-Brownian motion, offering new probabilistic representations and insights into the limiting behavior.

Findings

Established asymptotic properties of the skeleton process

Derived new probabilistic representations of the limiting extremal process

Provided insights into the driving mechanisms behind the extremal behavior

Abstract

Recently Ren et al. [Stoch. Proc. Appl., 137 (2021)] have proved that the extremal process of the super-Brownian motion converges in distribution in the limit of large times. Their techniques rely heavily on the study of the convergence of solutions to the Kolmogorov-Petrovsky-Piscounov equation along the lines of [M. Bramson, Mem. Amer. Math. Soc., 44 (1983)]. In this paper we take a different approach. Our approach is based on the skeleton decomposition of super-Brownian motion. The skeleton may be interpreted as immortal particles that determine the large time behaviour of the process. We exploit this fact and carry asymptotic properties from the skeleton over to the super-Brownian motion. Some new results concerning the probabilistic representations of the limiting process are obtained, which cannot be directly obtained through the results of [Y.-X. Ren et al., Stoch. Proc. Appl., 137 (2021)]. Apart from the results, our approach offers insights into the driving force behind the limiting process for super-Brownian motions.