Supercritical superprocesses: Proper normalization and non-degenerate strong limit

Yan Xia Ren, Renming Song, Rui Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

Suppose that X = {Xt, t ⩾ 0;ℙμ} is a supercritical superprocess in a locally compact separable metric space E. Let φ0 be a positive eigenfunction corresponding to the first eigenvalue λ0 of the generator of the mean semigroup of X. Then Mt:=e−λ0t〈ϕ0,Xt〉 is a positive martingale. Let M be the limit of Mt. It is known (see Liu et al. (2009)) that M is non-degenerate if and only if the L log L condition is satisfied. In this paper we are mainly interested in the case when the L log L condition is not satisfied. We prove that, under some conditions, there exist a positive function γt on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E, Mt:=e-λ0t〈φ0,Xt〉 We also give the almost sure limit of γt〈f, Xt〉 for a class of general test functions f.

Original languageEnglish (US)
Pages (from-to)1519-1552
Number of pages34
JournalScience China Mathematics
Volume62
Issue number8
DOIs
StatePublished - Aug 1 2019

Keywords

  • 60F15
  • 60J68
  • Seneta-Heyde norming
  • martingales
  • non-degenerate strong limit
  • superprocesses

ASJC Scopus subject areas

  • Mathematics(all)

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