## Abstract

Let X = {X_{t}, t ≥ 0; P_{μ}} be a critical superprocess starting from a finite measure μ. Under some conditions, we first prove that lim_{t→∞} tP_{μ}(∥X_{t}≠ = 0) = ν^{−1}〈φ_{0}, μ〉, where φ_{0} is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator L of the mean semigroup of X, and ν is a positive constant. Then we show that, for a large class of functions f, conditioning on ∥X_{t}≠ = 0, t^{−1}〈f, X_{t}〉 converges in distribution to 〈f, ψ_{0}〉_{m}W, where W is an exponential random variable, and ψ_{0} is the eigenfunction corresponding to the first eigenvalue of the dual of L. Finally, if 〈f, ψ_{0}〉_{m} = 0, we prove that, conditioning on ∥X_{t}∥ = 0, (t^{−1}〈φ_{0}, X_{t}〉, t^{−1/2}〈f, X_{t}〉) converges in distribution to (W, G(f) √ W), where G(f) ∼ N (0, σ^{2}_{f}) is a normal random variable, and W and G(f) are independent.

Original language | English (US) |
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Pages (from-to) | 235-276 |

Number of pages | 42 |

Journal | Illinois Journal of Mathematics |

Volume | 59 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2015 |

## ASJC Scopus subject areas

- General Mathematics