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

Zhen-Qing Chen, Yan-Xia Ren, Renming Song, Rui Zhang

Published 2015-02-05Version 1

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 $$ \psi(x,\lambda)=-a(x)\lambda+b(x)\lambda^2+\int_{(0,+\infty)}(e^{-\lambda y}-1+\lambda y)n(x,dy), \quad x\in E, \quad\lambda> 0, $$ where $a\in \mathcal{B}_b(E)$, $b\in \mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\infty)$ satisfying $$ \sup_{x\in E}\int_0^\infty y^2 n(x,dy)<\infty. $$ Put $T_tf(x)=\P_{\delta_x}\langle f,X_t\rangle$. Let $\lambda_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and $\phi_0$ and $\wh{\phi}_0$ 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 $\phi_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$ \lim_{t\rightarrow\infty}e^{-\lambda_0 t}\langle f, X_t\rangle = W_\infty\int_E\wh{\phi}_0(y)f(y)m(dy),\quad \P_{\mu}\mbox{-a.s.}, $$ for any finite initial measure $\mu$ on $E$ with compact support, where $W_\infty$ is the martingale limit defined by $W_\infty:=\lim_{t\to\infty}e^{-\lambda_0t}\langle \phi_0, X_t\rangle$. Moreover, the exceptional set in the above limit does not depend on the initial measure $\mu$ and the function $f$.