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

Research paper by **Zhen-Qing Chen, Yan-Xia Ren, Renming Song, Rui Zhang**

Indexed on: **08 Feb '15**Published on: **08 Feb '15**Published in: **Mathematics - Probability**

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#### Abstract

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
$T_tf(x)=\mathbb{P}_{\delta_x}< f,X_t>$. Let $\lambda_0>0$ be the largest
eigenvalue of the generator $L$ of $T_t$, and $\phi_0$ and $\hat{\phi}_0$ be
the eigenfunctions of $L$ and $\hat{L}$ (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 $$ \lim_{t\rightarrow\infty}e^{-\lambda_0 t}<
f, X_t> = W_\infty\int_E\hat{\phi}_0(y)f(y)m(dy),\quad \mathbb{P}_{\mu}{-a.s.},
$$ for any finite initial measure $\mu$ on $E$ with compact support, where
$W_\infty$ is the martingale limit defined by
$W_\infty:=\lim_{t\to\infty}e^{-\lambda_0t}< \phi_0, X_t>$. Moreover, the
exceptional set in the above limit does not depend on the initial measure $\mu$
and the function $f$.