Strong law of large numbers for supercritical superprocesses under second moment condition

Suppose that $X=\{X_t,t\ge 0\}$ is a supercritical superprocess on a locally compact separable metric space $(E,m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$\psi (x,\lambda )=-a(x)\lambda +b(x){\lambda }^2+{\int }_{(0,+\mathrm{\infty })}(e^{-\lambda y}-1+\lambda y)n(x,dy),x\in E,\lambda >0,$$ where $a\in $\mathcal{B}$_b(E)$,$b\in $\mathcal{B}$b^+(E)$and$n$isakernelfrom$E$to$(0,\infty)$satisfying$${\sup }_{x\in E}{\int }_0^{\mathrm{\infty }}{y}^{2}n(x,dy)<\mathrm{\infty }\mathrm{.}$$Put$T_tf(x)=$\mathbb{P}${\delta_x}< f,X_t>$\mathrm{.}Let$\lambda_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and ${\varphi }_0$ and ${\widehat{\varphi }}_0$ be the eigenfunctions of $L$ and $\widehat{L}$ (the dural of $L$) respectively associated with ${\lambda }_0$. Under some conditions on the spatial motion and the ${\varphi }_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$\underset{t\to \mathrm{\infty }}{\lim }{e}^{-{\lambda }_{0}t}<f,X_t>=W_{\mathrm{\infty }}{\int }_E{\widehat{\varphi }}_0(y)f(y)m(dy),{\mathbb{P}}_{\mu }-a\mathrm{.}s\mathrm{.},$$ for any finite initial measure $\mu$ on $E$ with compact support, where $W_{\mathrm{\infty }}$ is the martingale limit defined by $W_{\mathrm{\infty }}:={\lim }_{t\to \mathrm{\infty }}{e}^{-{\lambda }_{0}t}<{\varphi }_0,X_t>$. Moreover, the exceptional set in the above limit does not depend on the initial measure $\mu$ and the function $f$.