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arXiv · 1807.02837

Limit theorems for a class of critical superprocesses with stable branching

Yan-Xia Ren·Renming Song·Zhenyao Sun
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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 $E[e^{-u\mathbf z^{(\gamma_0-1)}}]=1-(1+u^{-(\gamma_0-1)})^{-1/(\gamma_0-1)}.$

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Yan-Xia Ren, Renming Song, Zhenyao Sun. 2018-07-08. Limit theorems for a class of critical superprocesses with stable branching. https://arxiv.org/abs/1807.02837

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