## 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.

Original language | English (US) |
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Pages (from-to) | 807-838 |

Number of pages | 32 |

Journal | Frontiers of Mathematics in China |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - Aug 5 2015 |

## Keywords

- Hunt process
- Superprocess
- h-transform
- martingale measure
- scaling limit theorem
- spectral gap

## ASJC Scopus subject areas

- Mathematics (miscellaneous)