Analytic birth-death processes: A Hilbert-space approach

Markus Kreer

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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
Issue number1
Publication statusPublished - Jan 1994


  • analytic semigroups
  • discreteness of spectrum
  • infinite tridiagonal matrices


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