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

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...

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Bibliographic Details
Published inJournal of applied probability Vol. 60; no. 3; pp. 737 - 764
Main Authors Chen, Anyue, Chen, Yong, Gao, Wu-Jun, Wu, Xiaohan
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.09.2023
Subjects
Branching (mathematics)
Decay
Eigenvalues
Field theory
Liouville equations
Lower bounds
Markov analysis
Markov processes
Mathematical functions
Operators (mathematics)
Original Article
Parameters
Partial differential equations
Transition probabilities
decay parameter
60J80
Quadratic branching process
Hardy-type inequality
generating functions
60J27
Sturm–Liouville operator
eigenvalue
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Summary: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.
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2022.91