Electronic Journal of Probability

On Dirichlet eigenvectors for neutral two-dimensional Markov chains

Nicolas Champagnat, Persi Diaconis, and Laurent Miclo

Full-text: Open access

Abstract

We consider a general class of discrete, two-dimensional Markov chains modeling the dynamics of a population with two types, without mutation or immigration, and neutral in the sense that type has no influence on each individual's birth or death parameters. We prove that all the eigenvectors of the corresponding transition matrix or infinitesimal generator $\Pi$ can be expressed as the product of ``universal'' polynomials of two variables, depending on each type's size but not on the specific transitions of the dynamics, and functions depending only on the total population size. These eigenvectors appear to be Dirichlet eigenvectors for $\Pi$ on the complement of triangular subdomains, and as a consequence the corresponding eigenvalues are ordered in a specific way. As an application, we study the quasistationary behavior of finite, nearly neutral, two-dimensional Markov chains, absorbed in the sense that $0$ is an absorbing state for each component of the process.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 63, 41 pp.

Dates
Accepted: 18 August 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062385

Digital Object Identifier
doi:10.1214/EJP.v17-1830

Mathematical Reviews number (MathSciNet)
MR2968670

Zentralblatt MATH identifier
1259.60078

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 39A14: Partial difference equations 47N30: Applications in probability theory and statistics 92D25: Population dynamics (general)

Keywords
Hahn polynomials two-dimensional difference equation neutral Markov chain multitype population dynamics Dirichlet eigenvector Dirichlet eigenvalue quasi-stationary distribution Yaglom limit coexistence

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Champagnat, Nicolas; Diaconis, Persi; Miclo, Laurent. On Dirichlet eigenvectors for neutral two-dimensional Markov chains. Electron. J. Probab. 17 (2012), paper no. 63, 41 pp. doi:10.1214/EJP.v17-1830. https://projecteuclid.org/euclid.ejp/1465062385


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References

  • Billiard, Sylvain; Tran, Viet Chi. A general stochastic model for sporophytic self-incompatibility. J. Math. Biol. 64 (2012), no. 1-2, 163–210.
  • Brezis, Haïm. Analyse fonctionnelle. (French) [Functional analysis] Théorie et applications. [Theory and applications] Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree] Masson, Paris, 1983. xiv+234 pp. ISBN: 2-225-77198-7
  • Cattiaux, Patrick; Collet, Pierre; Lambert, Amaury; Martènez, Servet; Méléard, Sylvie; San Martín, Jaime. Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37 (2009), no. 5, 1926–1969.
  • Cattiaux, Patrick; Méléard, Sylvie. Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction. J. Math. Biol. 60 (2010), no. 6, 797–829.
  • Cavender, James A. Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Probab. 10 (1978), no. 3, 570–586.
  • Champagnat, Nicolas; Lambert, Amaury. Evolution of discrete populations and the canonical diffusion of adaptive dynamics. Ann. Appl. Probab. 17 (2007), no. 1, 102–155.
  • Darroch, J. N.; Seneta, E. On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probability 2 1965 88–100.
  • Darroch, J. N.; Seneta, E. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probability 4 1967 192–196.
  • Dunkl, Charles F.; Xu, Yuan. Orthogonal polynomials of several variables. Encyclopedia of Mathematics and its Applications, 81. Cambridge University Press, Cambridge, 2001. xvi+390 pp. ISBN: 0-521-80043-9
  • Ferrari, Pablo A.; Martínez, Servet; Picco, Pierre. Existence of nontrivial quasi-stationary distributions in the birth-death chain. Adv. in Appl. Probab. 24 (1992), no. 4, 795–813.
  • Flaspohler, David C. Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Statist. Math. 26 (1974), 351–356.
  • Gantmacher, F. R. Matrizentheorie. (German) [Theory of matrices] With a foreword by D. P. Želobenko. Translated from the Russian by Helmut Boseck, Dietmar Soyka and Klaus Stengert. Hochschulbücher für Mathematik [University Books for Mathematics], 86. VEB Deutscher Verlag der Wissenschaften, Berlin, 1986. 654 pp. ISBN: 3-326-00001-4
  • Gosselin, Frédéric. Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology. Ann. Appl. Probab. 11 (2001), no. 1, 261–284.
  • Högnäs, Göran. On the quasi-stationary distribution of a stochastic Ricker model. Stochastic Process. Appl. 70 (1997), no. 2, 243–263.
  • Karlin, S.; McGregor, J. Linear growth models with many types and multidimensional Hahn polynomials. Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 261–288. Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975.
  • Kesten, Harry. A ratio limit theorem for (sub) Markov chains on $\{1,2,\cdots\}$ with bounded jumps. Adv. in Appl. Probab. 27 (1995), no. 3, 652–691.
  • Khare, Kshitij; Zhou, Hua. Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions. Ann. Appl. Probab. 19 (2009), no. 2, 737–777.
  • Kijima, Masaaki; Seneta, E. Some results for quasi-stationary distributions of birth-death processes. J. Appl. Probab. 28 (1991), no. 3, 503–511.
  • M. Kimura, The neutral theory of molecular evolution, Cambridge University Press, 1983.
  • Motoo Kimura, Solution of a process of random genetic drift with a continuous model, Proc. Nat. Acad. Sci. 41 (1955), 144–150.
  • Murray, J. D. Mathematical biology. Second edition. Biomathematics, 19. Springer-Verlag, Berlin, 1993. xiv+767 pp. ISBN: 3-540-57204-X
  • Nåsell, Ingemar. On the quasi-stationary distribution of the stochastic logistic epidemic. Epidemiology, cellular automata, and evolution (Sofia, 1997). Math. Biosci. 156 (1999), no. 1-2, 21–40.
  • P. K. Pollett, Quasi-stationary distributions: A bibliography, available on http://www.maths.uq.edu.au/erb+ +pkp/papers/qsds/, 2011.
  • Seneta, E.; Vere-Jones, D. On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probability 3 1966 403–434.
  • van Doorn, Erik A. Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. in Appl. Probab. 23 (1991), no. 4, 683–700.
  • Zettl, Anton. Sturm-Liouville theory. Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. xii+328 pp. ISBN: 0-8218-3905-5