Search

In this paper we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom's exponential limit law for critical superprocesses. [ABSTRACT FROM AUTHOR]

Consider a supercritical superprocess X = { X, 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 where $$a \in B_b (E)$$, $$b \in B_b^ + (E)$$, and n is a kernel from E to (0,+∞) satisfying sup∫ y n( x, d y) < +∞. Put $$T_t f(x) = \mathbb{P}_{\delta _x } \left\langle {f,X_t } \right\rangle$$. Suppose that the semigroup { T; t ⩾ 0} is compact. Let λ be the eigenvalue of the (possibly non-symmetric) generator L of { T} that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let ϕ and $$\hat \varphi _0$$ be the eigenfunctions of L and $$\hat L$$ (the dual of L) associated with λ, respectively. Assume λ > 0. Under some conditions on the spatial motion and the ϕ-transform of the semigroup { T}, we prove that for a large class of suitable functions f, 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. [ABSTRACT FROM AUTHOR]