## Abstract

Consider the σ-finite measure-valued diffusion corresponding to the evolution equation u_{t} = Lu + β(x)u - f(x, u), where f(x,u) = α(x)u^{2} + ∫_{0}^{∞} (e^{-ku} - 1 + ku)n(x,dk) and n is a smooth kernel satisfying an integrability condition. We assume that β, α ∈ C^{η}(ℝ^{d}) with η ∈ (0, 1], and α > 0. Under appropriate spectral theoretical assumptions we prove the existence of the random measure lim_{t↑∞} e^{-λct}X_{t}(dx) (with respect to the vague topology), where λ_{c} is the generalized principal eigenvalue of L + β on ℝ^{d} and it is assumed to be finite and positive, completing a result of Pinsky on the expectation of the rescaled process. Moreover, we prove that this limiting random measure is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator L + β. When β is bounded from above, X is finite measure-valued. In this case, under an additional assumption on L + β, we can actually prove the existence of the previous limit with respect to the weak topology. As a particular case, we show that if L corresponds to a positive recurrent diffusion Y and β is a positive constant, then lim_{t↑∞} e^{-βt} X_{t}(dx) exists and equals a nonnegative nondegenerate random multiple of the invariant measure for Y. Taking L = 1/2 δ on ℝ and replacing β by δ_{0} (super-Brownian motion with a single point source), we prove a similar result with λ_{c} replaced by 1/2 and with the deterministic measure e^{-|x|}dx, giving an answer in the affirmative to a problem proposed by Engländer and Fleischmann [Stochastic Process. Appl. 88 (2000) 37-58]. The proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups.

Original language | English |
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Pages (from-to) | 683-722 |

Number of pages | 40 |

Journal | Annals of Probability |

Volume | 30 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2002 |

Externally published | Yes |

## Keywords

- Invariant curve
- Measure-valued process
- Scaling limit
- Single point source
- Super-Brownian motion
- Superprocess

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty