Limit theorems for some critical superprocesses

Yan Xia Ren, Renming Song, Rui Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

Let X = {Xt, t ≥ 0; Pμ} be a critical superprocess starting from a finite measure μ. Under some conditions, we first prove that limt→∞ tPμ(∥Xt≠ = 0) = ν−1〈φ0, μ〉, where φ0 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 ∥Xt≠ = 0, t−1〈f, Xt〉 converges in distribution to 〈f, ψ0mW, where W is an exponential random variable, and ψ0 is the eigenfunction corresponding to the first eigenvalue of the dual of L. Finally, if 〈f, ψ0m = 0, we prove that, conditioning on ∥Xt∥ = 0, (t−1〈φ0, Xt〉, t−1/2〈f, Xt〉) converges in distribution to (W, G(f) √ W), where G(f) ∼ N (0, σ2f) is a normal random variable, and W and G(f) are independent.

Original languageEnglish (US)
Pages (from-to)235-276
Number of pages42
JournalIllinois Journal of Mathematics
Volume59
Issue number1
DOIs
StatePublished - Mar 1 2015

ASJC Scopus subject areas

  • Mathematics(all)

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