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Asymptotic moments of spatial branching processes
Language
en
Article de revue
This item was published in
Probability Theory and Related Fields. 2022-04-25, vol. 184, p. 805-858
Springer Verlag
English 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 ...Read more >
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.Read less <
English Keywords
Asymptotic behaviour
Non-local branching
Superprocesses
Branching processes
Moments
Origin
Hal imported