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

Abstract

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 +∞ y 2 n(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 λ 0 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 ϕ 0 and $$\hat \varphi _0$$ be the eigenfunctions of L and $$\hat L$$ (the dual of L) associated with λ 0, respectively. Assume λ 0 > 0. Under some conditions on the spatial motion and the ϕ 0-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.