Analytic birth-death processes: A Hilbert-space approach

Markus Kreer

Research output: Contribution to journalArticle

Abstract

Methods of Hilbert space theory together with the theory of analytic semigroups lead to an alternative approach for discussing an analytic birth and death process with the backward equations {greater-than with dot}k = ?k - 1gk - 1 - (µk+?k)gkk+1, k = 0, 1, 2, ..., where ?- 1 = 0 = µ0. For rational growing forward and backward transition rates ?k = O(k?), µk = O(k?) (as k ? 8), with 0 < ? < 1, the existence and uniqueness of a solution (which is analytic for t > 0) can be proved under fairly general conditions; so can the discreteness of the spectrum. Even in the critical case of asymptotically symmetric transition rates ?k ~ µk ~ k? one obtains for rational growing transition rates with 0 < ? < 1 discreteness of the spectrum, generalizing a result of Chihara (1987) and disproving the traditional belief in a continuous spectrum. © 1994.

Original languageEnglish
Pages (from-to)65-74
Number of pages10
JournalStochastic Processes and their Applications
Volume49
Issue number1
Publication statusPublished - Jan 1994

Keywords

  • analytic semigroups
  • discreteness of spectrum
  • infinite tridiagonal matrices

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