Spine decompositions and limit theorems for a class of critical superprocesses

TL;DR

This paper develops spine decomposition theorems for critical superprocesses, providing probabilistic proofs for their asymptotic survival behavior and Yaglom's law, advancing understanding of their long-term dynamics.

Contribution

It introduces new spine decomposition theorems for critical superprocesses based on size-biased Poisson measures, offering novel probabilistic proofs of their asymptotic properties.

Findings

Proved a decomposition theorem for size-biased Poisson measures.

Established spine and 2-spine decomposition theorems for critical superprocesses.

Provided probabilistic proofs of survival probability asymptotics and Yaglom's law.

Abstract

In this paper, we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom's exponential limit law for critical superprocesses.