Branching diffusions, superdiffusions and random media

TL;DR

This paper introduces spatial branching processes and superprocesses, exploring their connections to nonlinear PDEs and their behavior in deterministic and random media, including classical and recent models in probability theory.

Contribution

It provides a comprehensive introduction to spatial branching processes and superprocesses, emphasizing their applications in deterministic environments and random media, bridging classical and modern research.

Findings

Connections between superprocesses and nonlinear PDEs

Analysis of branching processes in random media

Overview of classical and recent models in probability

Abstract

Spatial branching processes became increasingly popular in the past decades, not only because of their obvious connection to biology, but also because superprocesses are intimately related to nonlinear partial differential equations. Another hot topic in today's research in probability theory is `random media', including the now classical problems on `Brownian motion among obstacles' and the more recent `random walks in random environment' and `catalytic branching' models. These notes aim to give a gentle introduction into some topics in spatial branching processes and superprocesses in deterministic environments (sections 2-6) and in random media (sections 7-11).