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 $ψ(x,λ)=-a(x)λ+b(x)λ^2+\int_{(0,+\infty)}(e^{-λy}-1+λy)n(x,dy), \quad x\in E, \quadλ> 0,$ 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 $T_tf(x)=\mathbb{P}_{δ_x}< f,X_t>$. Let $λ_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and $ϕ_0$ and $\hatϕ_0$ be the eigenfunctions of $L$ and $\hat{L}$ (the dural of $L$) respectively associated with $λ_0$. Under some conditions on the spatial motion and the $ϕ_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $\lim_{t\rightarrow\infty}e^{-λ_0 t}< f, X_t> = W_\infty\int_E\hatϕ_0(y)f(y)m(dy),\quad \mathbb{P}_μ{-a.s.},$ for any finite initial measure $μ$ on $E$ with compact support, where $W_\infty$ is the martingale limit defined by $W_\infty:=\lim_{t\to\infty}e^{-λ_0t}< ϕ_0, X_t>$. Moreover, the exceptional set in the above limit does not depend on the initial measure $μ$ and the function $f$.

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