Limit theorems for a class of critical superprocesses with stable branching

TL;DR

This paper studies the long-term behavior of a class of critical superprocesses with stable branching, showing convergence properties and describing the limiting distribution after rescaling.

Contribution

It establishes new limit theorems for critical superprocesses with stable branching mechanisms and describes the asymptotic distribution of the process conditioned on survival.

Findings

Survival probability converges to zero and is regularly varying with a specific index.

Rescaled distributions conditioned on non-extinction converge weakly to a non-degenerate limit.

Explicit Laplace transform of the limiting distribution is derived.

Abstract

We consider a critical superprocess $\{X;\mathbf P_\mu\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index $\gamma_0 > 1$. We first show that, under some conditions, $\mathbf P_{\mu}(\|X_t\|\neq 0)$ converges to $0$ as $t\to \infty$ and is regularly varying with index $(\gamma_0-1)^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t(f);\mathbf P_\mu(\cdot|\|X_t\|\neq 0)\}$, after appropriate rescaling, converges weakly to a positive random variable $\mathbf z^{(\gamma_0-1)}$ with Laplace transform $E[e^{-u\mathbf z^{(\gamma_0-1)}}]=1-(1+u^{-(\gamma_0-1)})^{-1/(\gamma_0-1)}.$