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On the theory of birth, death and diffusion processes

Published online by Cambridge University Press:  14 July 2016

A. W. Davis
A. W. Davis*
Affiliation:
Australian National University
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Extract

Several authors have recently discussed the asymptotic properties of stochastic populations which diffuse randomly throughout a given region. Sevast'yanov ([8], [9]) has investigated the extinction probability of a Markovian population in a compact region with an absorbing boundary, his analysis being in terms of “generation times”. Adke and Moyal have considered the spatial dispersion of a population which multiplies according to a simple time-dependent birth-and-death process and undergoes Gaussian diffusion on the real line ([2] and [3]) or on a finite interval with reflecting boundaries [1]. A serious limitation in Adke and Moyal's asymptotic results is that they are conditional upon a finite number of survivors. Moyal [7] has also obtained some basic formulae for a Markovian population diffusing through a general space.

Type
Research Papers
Information
Journal of Applied Probability , Volume 2 , Issue 2 , December 1965 , pp. 293 - 322
Copyright
Copyright © Applied Probability Trust 

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References

[1] Adke, S. R. (1964) A stochastic population diffusing on a finite interval. J. Ind. Statist. Assoc. 2, 32–40.Google Scholar
[2] Adke, S. R. The generalized birth and death process and Gaussian diffusion. To appear in J. Math. Anal. Appl. Google Scholar
[3] Adke, S. R. and Moyal, J. E. (1963) A birth, death and diffusion process. J. Math. Anal Appl. 7, 209–224.CrossRefGoogle Scholar
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[8] Sevast'Yanov, B. A. (1958) Branching stochastic processes for particles diffusing in a bounded domain with absorbing boundaries. Theor. Prob. Appl. 3, 111–126.Google Scholar
[9] Sevast'Yanov, B. A. (1961) Extinction conditions for branching stochastic processes with diffusion. Theor. Prob. Appl. 6, 253–263.Google Scholar