Strong law of large number of a class of super-diffusions

In this paper we prove that, under certain conditions, a strong law of large numbers holds for a class of super-diffusions $X$ corresponding to the evolution equation ${\mathrm{\partial }}_t{u}_t=L{u}_t+\beta u_t-\psi (u_t)$ on a bounded domain $D$ in ${\mathbb{R}}^d$, where $L$ is the generator of the underlying diffusion and the branching mechanism $\psi(x,\lambda)=1/2\alpha(x)\lambda^2+\int_0^\infty (e^{-\lambda r}-1+\lambda r)n(x, {\rm d}r)$satisfies$\sup_{x\in D}\int_0^\infty (r\wedge r^2) n(x,{\rm d}r)<\infty$.