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Journal Articles Probability Theory and Related Fields Year : 2022

Asymptotic moments of spatial branching processes

Abstract

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. The method we employ is extremely robust and we are able to extract similarly precise results that additionally give us the moment growth with time of \int_0^t ⟨g, X_t⟩ds, for bounded measurable g on E.
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Dates and versions

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

Identifiers

Cite

Isaac Gonzalez, Emma Horton, Andreas E Kyprianou. Asymptotic moments of spatial branching processes. Probability Theory and Related Fields, 2022, 184, pp.805-858. ⟨10.1007/s00440-022-01131-2⟩. ⟨hal-03321803v2⟩
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