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Sturm–Liouville theory and decay parameter for quadratic markov branching processes

Part of: Markov processes

Published online by Cambridge University Press:  19 January 2023

Anyue Chen ,
Yong Chen Open the ORCID record for Yong Chen [Opens in a new window] ,
Wu-Jun Gao  and
Xiaohan Wu
Anyue Chen*
Affiliation:
Southern University of Science and Technology and University of Liverpool
Yong Chen*
Affiliation:
Jiangxi Normal University
Wu-Jun Gao*
Affiliation:
Shenzhen Technology University
Xiaohan Wu*
Affiliation:
Harbin Institute of Technology
*
*Postal address: Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China; Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK. Email address: achen@liv.ac.uk
**Postal address: School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, China. Email address: zhishi@pku.org.cn
***Postal address: College of Big Data and Internet, Shenzhen Technology University, Shenzhen, 518118, China. Email address: gaowujun@sztu.edu.cn
****Postal address: Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China. Email address: 11849455@mail.sustech.edu.cn
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Abstract

For a quadratic Markov branching process (QMBP), we show that the decay parameter is equal to the first eigenvalue of a Sturm–Liouville operator associated with the partial differential equation that the generating function of the transition probability satisfies. The proof is based on the spectral properties of the Sturm–Liouville operator. Both the upper and lower bounds of the decay parameter are given explicitly by means of a version of Hardy’s inequality. Two examples are provided to illustrate our results. The important quantity, the Hardy index, which is closely linked to the decay parameter of the QMBP, is deeply investigated and estimated.

Keywords

Quadratic branching processdecay parameterSturm–Liouville operatorHardy-type inequalitygenerating functionseigenvalue

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 737 - 764
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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