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

Consider a supercritical superprocess X={X_t, t ⩾ 0} 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 $$\psi(x,\lambda)=-a(x)\lambda+b(x)\lambda^2+\int_{(0,+\infty)} {(e^{-\lambda y}-1 +\lambda y)} n(x,dy),x\in E,\lambda > 0,$$ where $$a \in B_b (E)$$, $$b \in B_b^ + (E)$$, and n is a kernel from E to (0,+∞) satisfying sup_x∈E∫₀^+∞y²n(x, dy) < +∞. Put $$T_t f(x)=\mathbb{P}_{\delta _x } \left\langle {f,X_t } \right\rangle$$. Suppose that the semigroup {T_t; t ⩾ 0} is compact. Let λ₀ be the eigenvalue of the (possibly non-symmetric) generator L of {T_t} that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let ϕ₀ and $$\hat \varphi _0$$ be the eigenfunctions of L and $$\hat L$$ (the dual of L) associated with λ₀, respectively. Assume λ₀ > 0. Under some conditions on the spatial motion and the ϕ₀-transform of the semigroup {T_t}, we prove that for a large class of suitable functions f, $$\mathop {\lim }\limits_{t \to + \infty } e^{ - \lambda _0 t} \left\langle {f,X_t } \right\rangle = W_\infty \int_E {\hat \varphi _0 (y)f(y)m(dy), \mathbb{P}_\mu - a.s.,}$$ for any finite initial measure µ on E with compact support, where W_∞ is the martingale limit defined by $$W_\infty :=\lim _{t \to + \infty } e^{ - \lambda _0 t} \left\langle {\varphi _0 ,X_t } \right\rangle$$. Moreover, the exceptional set in the above limit does not depend on the initial measure µ and the function f.