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.