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 }^{\mathrm{2}}+{\displaystyle {\int }_{(\mathrm{0},+\infty )}({\mathrm{e}}^{-\lambda y}-\mathrm{1}+\lambda y)n(x,\mathrm{d}y),}\hspace{1em}x\in E,\lambda >\mathrm{0},

where a\in {{B}}_b(E),b\in {{B}}_b^{+}(E), and n is a kernel from E to (0,+∞) satisfying . Put T_{t}f(x)={\mathbb{P}}_{{\delta }_x}\langle f,X_t\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 {\varphi }_{\mathrm{0}} and {\widehat{\varphi }}_{\mathrm{0}} be the eigenfunctions of L and \widehat{L}(the dual of L) associated with λ₀, respectively. Assume λ₀>0. Under some conditions on the spatial motion and the {\varphi }_{\mathrm{0}}-transform of the semigroup {T_t}, we prove that for a large class of suitable functions f,

\underset{t\to +\infty }{\lim }{\mathrm{e}}^{-{\lambda }_{\mathrm{0}}t}\langle f,X_t\rangle =W_{\infty }{\displaystyle {\int }_E{\widehat{\varphi }}_{\mathrm{0}}(y)f(y)m(\mathrm{d}y)},\hspace{1em}{\mathbb{P}}_{\mu }-\mathrm{a}.\mathrm{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 }{\mathrm{e}}^{-{\lambda }_{\mathrm{0}}t}\langle {\varphi }_{\mathrm{0}},X_t\rangle. Moreover, the exceptional set in the above limit does not depend on the initial measure μ and the function f.

Key words： Superprocess; scaling limit theorem; Hunt process; spectral gap; h-transform; martingale measure

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