## Abstract

Consider a superdiffusion X on ℝ^{d} corresponding to the semi-linear operator A(u) = Lu + βu-ku^{2}, where L is a second order elliptic operator, β(.) is in the Kato class, and k(.) ≥ 0 is bounded on compact subsets of ℝ^{d} 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 ||X_{t}||, the total mass of X_{t}, is λ. We also describe the limiting behavior of exp(-λt)||X_{t}||, as t →. This should be compared to the result in (Ann. Probab. 32 (2004) 78-99), which says that the generalized principal eigenvalue λ_{2} 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 ≥ λ_{2}, and we discuss cases when λ > λ_{2} and when λ = λ_{2}. 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 λ_{2} only, and it turns out to be equivalent to local extinction.)

Original language | English (US) |
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Pages (from-to) | 448-482 |

Number of pages | 35 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 52 |

Issue number | 1 |

DOIs | |

State | Published - 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