Asymptotic moments of spatial branching processes
Asymptotic moments of spatial branching processes
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DOI：
10.1007/s00440-022-01131-2
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发表时间：
2022-04
影响因子：
2
通讯作者：
I. González;E. Horton;A. Kyprianou
中科院分区：
文献类型：
--
作者：
I. González;E. Horton;A. Kyprianou
Suppose thatis either a superprocess or a branching Markov process on a general spaceE, with non-local branching mechanism and probabilities, when issued from a unit mass at. For a general setting in which the first moment semigroup ofXdisplays a Perron–Frobenius type behaviour, we show that, forand any positive bounded measurable functionfonE, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{t\rightarrow \infty } g_k(t){\mathbb {E}}_{\delta _x}[\langle f, X_t\rangle ^k] = C_k(x, f), \end{aligned}$$\end{document}where the constantcan be identified in terms of the principal right eigenfunction and left eigenmeasure andis an appropriate deterministic normalisation, which can be identified explicitly as either polynomial intor exponential int, depending on whetherXis 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, for bounded measurablefonE.
Suppose thatis either a superprocess or a branching Markov process on a general spaceE, with non-local branching mechanism and probabilities, when issued from a unit mass at. For a general setting in which the first moment semigroup ofXdisplays a Perron–Frobenius type behaviour, we show that, forand any positive bounded measurable functionfonE, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{t\rightarrow \infty } g_k(t){\mathbb {E}}_{\delta _x}[\langle f, X_t\rangle ^k] = C_k(x, f), \end{aligned}$$\end{document}where the constantcan be identified in terms of the principal right eigenfunction and left eigenmeasure andis an appropriate deterministic normalisation, which can be identified explicitly as either polynomial intor exponential int, depending on whetherXis 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, for bounded measurablefonE.