GONZALEZ, Isaac; HORTON, Emma; KYPRIANOU, Andreas

doi:10.1007/s00440-022-01131-2

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GONZALEZ, Isaac
Department of Mathematical Sciences [Bath]

HORTON, Emma
Méthodes avancées d’apprentissage statistique et de contrôle [ASTRAL]

KYPRIANOU, Andreas
Department of Mathematical Sciences [Bath]

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Probability Theory and Related Fields. 2022-04-25, vol. 184, p. 805-858

Springer Verlag

Résumé en anglais

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.< Réduire

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Asymptotic behaviour

Non-local branching

Superprocesses

Branching processes

Moments

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Asymptotic moments of spatial branching processes

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