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Pré-Publication, Document De Travail Année : 2021

Asymptotic moments of spatial branching processes

Résumé

Suppose that X = (X t , t ≥ 0) is either a superprocess or a branching Markov process on a general space E, with non-local branching mechanism and probabilities P δx , when issued from a unit mass at x ∈ E. For a general setting in which the first moment semigroup of X displays a Perron-Frobenius type behaviour, we show that, for k ≥ 2 and any positive bounded measurable function f on E, lim t→∞ g(t)E δx [ f, X t k ] = C k (x, f) where the constant C k (x, f) can be identified in terms of the principal right eigenfunction and left eigen-measure and g(t) is an appropriate determinisitic normalisation, which can be identified explicitly as either polynomial in t or exponential in t, depending on whether X is a critical, supercritical or subcritical process.
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Dates et versions

hal-03321803 , version 1 (18-08-2021)
hal-03321803 , version 2 (20-12-2021)

Identifiants

  • HAL Id : hal-03321803 , version 1

Citer

Isaac Gonzalez, Emma Horton, Andreas E Kyprianou. Asymptotic moments of spatial branching processes. 2021. ⟨hal-03321803v1⟩
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