The Annals of Applied Probability

A transition function expansion for a diffusion model with selection

A. D. Barbour, S. N. Ethier, and R. C. Griffiths

Full-text: Open access


Using duality, an expansion is found for the transition function of the reversible $K$-allele diffusion model in population genetics. In the neutral case, the expansion is explicit but already known. When selection is present, it depends on the distribution at time $t$ of a specified $K$-type birth-and-death process starting at “infinity.” The latter process is constructed by means of a coupling argument and characterized as the Ray process corresponding to the Ray–Knight compactification of the $K$-dimensional nonnegative-integer lattice.

Article information

Ann. Appl. Probab., Volume 10, Number 1 (2000), 123-162.

First available in Project Euclid: 25 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65] 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 92D10: Genetics {For genetic algebras, see 17D92}

Finite-dimensional diffusion process population genetics duality reversibility multitype birth-and-death process coupling Ray-Knight compactification


Barbour, A. D.; Ethier, S. N.; Griffiths, R. C. A transition function expansion for a diffusion model with selection. Ann. Appl. Probab. 10 (2000), no. 1, 123--162. doi:10.1214/aoap/1019737667.

Export citation


  • Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming-Viot measurevalued diffusion. Ann.Probab.24 698-742.
  • Donnelly, P. and Kurtz, T. G. (1999). Genealogical processes for Fleming-Viot models with selection and recombination. Ann.Appl.Probab.To appear.
  • Ethier, S. N. and Griffiths, R. C. (1993). The transition function of a Fleming-Viot process. Ann. Probab. 21 1571-1590.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Griffiths, R. C. (1979). A transition density expansion for a multi-allele diffusion model. Appl.Probab.11 310-325.
  • Griffiths, R. C. (1980). Lines of descent in the diffusion approximation of neutral Wright-Fisher models. Theoret.Population Biol.17 37-50.
  • Griffiths, R. C. and Li, W.-H. (1983). Simulating allele frequencies in a population and the genetic differentiation of populations under mutation pressure. Theoret.Population Biol. 32 19-33.
  • Kimura, M. (1955). Solution of a process of random genetic drift with a continuous model. Proc. Nat.Acad.Sci.U.S.A.41 144-150.
  • Kimura, M. (1957). Some problems of stochastic processes in genetics. Ann.Math.Statist.28 882-901.
  • Kingman, J. F. C. (1982). The coalescent. Stochastic Process.Appl.13 235-248.
  • Krone, S. M. and Neuhauser, C. (1997). Ancestral processes with selection. Theoret.Population Biol. 51 210-237.
  • Li, Z., Shiga, T. and Yao, L. (1999). A reversibility problem for Fleming-Viot diffusion processes. Electronic Comm.Probab.4 65-76.
  • Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.
  • Overbeck, L. and R ¨ockner, M. (1997). Geometric aspects of finite-and infinite-dimensional Fleming-Viot processes. Random Oper.Stochastic Equations 5 35-58.
  • Sato, K. (1978). Diffusion operators in population genetics and convergence of Markov chains. Measure Theory Applications to Stochastic Analysis.Lecture Notes in Math.695 127- 137. Springer, Berlin.
  • Shiga, T. (1981). Diffusion processes in population genetics. J.Math.Kyoto Univ.21 133-151.
  • Shimakura, N. (1977). Equations diff´erentielles provenant de la g´en´etique des populations. T ohoku Math.J.29 287-318.
  • Tavar´e, S. (1984). Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret.Population Biol.26 119-164.
  • Williams, D. (1979). Diffusions, Markov Processes, and Martingales 1. Wiley, Chichester.
  • Wright, S. (1949). Adaptation and selection. In Genetics, Paleontology, and Evolution (G. L. Jepson, E. Mayr and G. G. Simpson, eds.) 365-389. Princeton Univ. Press.
  • Xia, A. (1994). Weak convergence of Markov processes with extended generators. Ann.Probab. 22 2183-2202.