Weak extinction versus global exponential growth of total mass for superdiffusions

János Engländer, Yan Xia Ren, Renming Song

Research output: Contribution to journalArticlepeer-review

Abstract

Consider a superdiffusion X on ℝd corresponding to the semi-linear operator A(u) = Lu + βu-ku2, where L is a second order elliptic operator, β(.) is in the Kato class, and k(.) ≥ 0 is bounded on compact subsets of ℝd and is positive on a set of positive Lebesgue measure. The main purpose of this paper is to complement the results obtained in (Ann. Probab. 32 (2004) 78.99), in the following sense. Let λ be the L-growth bound of the semigroup corresponding to the Schrodinger-type operator L + β. If λ ≠ 0, then we prove that, in some sense, the exponential growth/decay rate of ||Xt||, the total mass of Xt, is λ. We also describe the limiting behavior of exp(-λt)||Xt||, as t →. This should be compared to the result in (Ann. Probab. 32 (2004) 78-99), which says that the generalized principal eigenvalue λ2 of the operator gives the rate of local growth when it is positive, and implies local extinction otherwise. It is easy to show that λ ≥ λ2, and we discuss cases when λ > λ2 and when λ = λ2. When λ = 0, and under some conditions on β, we give a necessary and sufficient condition for the superdiffusion X to exhibit weak extinction. We show that the branching intensity k affects weak extinction; this should be compared to the known result that k does not affect weak local extinction. (The latter depends on the sign of λ2 only, and it turns out to be equivalent to local extinction.)

Original languageEnglish (US)
Pages (from-to)448-482
Number of pages35
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume52
Issue number1
DOIs
StatePublished - Feb 2016

Keywords

  • Gauge theorem
  • Growth bound
  • H-transform
  • Kato class
  • Measure-valued process
  • Principal eigenvalue
  • Superdiffusion
  • Superprocess
  • Total mass
  • Weak extinction

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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