Limit theorems for a class of critical superprocesses with stable branching

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, ${\mathbf{P}}_{\mu }(\mathrm{\parallel }{X}_t\mathrm{\parallel }\mathrm{\ne }0)$ converges to $0$ as $t\to \mathrm{\infty }$ 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 $\{X_t(f);{\mathbf{P}}_{\mu }(\cdot \mathrm{\mid }\mathrm{\parallel }{X}_t\mathrm{\parallel }\mathrm{\ne }0)\}$, after appropriate rescaling, converges weakly to a positive random variable ${\mathbf{z}}^{({\gamma }_0-1)}$ with Laplace transform $E[e^{-u\mathbf z^{(\gamma_0-1)}}]=1-(1+u^{-(\gamma_0-1)})^{-1/(\gamma_0-1)}.$