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Applications of Feller-Reuter-Riley transition functions

Applications of Feller-Reuter-Riley transition functions

Chen, Anyue (2001) Applications of Feller-Reuter-Riley transition functions. Journal of Mathematical Analysis and Applications, 260 (2). pp. 439-456. ISSN 0022-247X (doi:10.1006/jmaa.2000.7466)

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A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.

Item Type: Article
Uncontrolled Keywords: Feller–Reuter–Riley functions, stable q-matrix, unstable q-matrix, Markov branching process, dual, Feller minimal q-functions, Williams-matrix
Subjects: Q Science > QA Mathematics
Pre-2014 Departments: School of Computing & Mathematical Sciences
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Last Modified: 14 Oct 2016 09:00
Selected for GREAT 2016: None
Selected for GREAT 2017: None
Selected for GREAT 2018: None

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