Asymptotic moments of spatial branching processes

• 成果类型: Article
• 署名作者: Gonzalez, Isaac; Horton, Emma; Kyprianou, Andreas E.
• 署名单位: University of Bath; Inria; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01131-2
• 发表日期: 2022
• 页码: 805-858
• 关键词: large numbers; stochastic methods; STRONG LAW; superprocesses

摘要

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-delta x, when issued from a unit mass at x is an element of 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 ->infinity) gk(t)E-delta x[< 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 eigenmeasure and g(k)(t) is an appropriate deterministic 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 integral(t)(0) < f, X-t > ds, for bounded measurable f on E.