Infinitely Divisible Random Measures and Superprocesses

TL;DR: In this article, the authors introduce the concept of local spatial clumping with a set of informal calculations that lead to the prediction that the continuous limit of branching particle systems in dimensions d ≥ 3 will lead to infinitely divisible random measures which are almost surely singular.

Abstract: The objective of these lectures is to serve as an introduction to the theory of measure-valued branching processes or super processes.This class of processes first arose from the study of continuous state branching in the work of Jirina (1958, 1964) and (1968). It was also linked to the study of stochastic evolution equations in (1975). In this introduction we look at two roots of this subject, namely, spatially distributed birth and death particle systems and stochastic partial differential equations with non-negative solutions. In Section 2 we carry out some exploratory calculations concerning the continuous limit of branching particle systems and their relation to stochastic partial differential equations. In addition, we introduce the ideas of local spatial clumping with a set of informal calculations that lead to the prediction that the continuum limit of branching particle systems in dimensions d≥3 will lead to infinitely divisible random measures which are almost surely singular.