Rocky Mountain Journal of Mathematics

Substitution Markov chains and Martin boundaries

David Koslicki and Manfred Denker

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Abstract

Substitution Markov chains have been introduced \cite {KoslickiThesis2012} as a new model to describe molecular evolution. In this note, we study the associated Martin boundaries from a probabilistic and topological viewpoint. An example is given that, although having a boundary homeomorphic to the well-known coin tossing process, has a metric description that differs significantly.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 6 (2016), 1963-1985.

Dates
First available in Project Euclid: 4 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1483520434

Digital Object Identifier
doi:10.1216/RMJ-2016-46-6-1963

Mathematical Reviews number (MathSciNet)
MR3591268

Zentralblatt MATH identifier
06673139

Subjects
Primary: 28D05: Measure-preserving transformations 31C35: Martin boundary theory [See also 60J50] 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 60J50: Boundary theory

Keywords
Substitution Markov chain Martin boundary random substitution

Citation

Koslicki, David; Denker, Manfred. Substitution Markov chains and Martin boundaries. Rocky Mountain J. Math. 46 (2016), no. 6, 1963--1985. doi:10.1216/RMJ-2016-46-6-1963. https://projecteuclid.org/euclid.rmjm/1483520434


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References

  • G. Alsmeyer and U. Rösler, The Martin entrance boundary of the Galton-Watson process, Ann. Inst. Poincar e Prob. Stat. 42 (2006), 591–606.
    Mathematical Reviews (MathSciNet): MR2259977
    Zentralblatt MATH: 1113.60089
    Digital Object Identifier: doi:10.1016/j.anihpb.2005.05.003
  • K.B. Athreya and P.E. Ney, Branching processes, Springer-Verlag, Berlin, 1972.
    Mathematical Reviews (MathSciNet): MR373040
    Zentralblatt MATH: 0259.60002
  • E.B. Dynkin, Boundary theory of Markov processes $($the discrete case$)$, Uspek. Mat. Nauk. 24 (1969), 1–42.
    Mathematical Reviews (MathSciNet): MR245096
    Zentralblatt MATH: 0222.60048
  • S. Evans, R. Grüubler and A. Wakolbinger, Trickle-down processes and their boundaries, Electr. J. Prob. 17 (2012), 58 pages.
    Mathematical Reviews (MathSciNet): MR2869248
    Zentralblatt MATH: 1246.60100
    Digital Object Identifier: doi:10.1214/EJP.v17-1698
  • G.A. Hunt, Markoff chains and Martin boundaries, Illinois J. Math. 4 (1960), 313–340.
    Mathematical Reviews (MathSciNet): MR123364
    Zentralblatt MATH: 0094.32103
    Project Euclid: euclid.ijm/1255456049
  • J.G. Kemeny, J.L. Snell and A.W. Knapp, Denumerable Markov chains, 2nd edition, Springer-Verlag, New York, 1965.
    Mathematical Reviews (MathSciNet): MR407981
    Zentralblatt MATH: 0178.53306
  • David Koslicki, Substitution Markov chains with applications to molecular evolution, Ph.D. dissertation, Pennsylvania State University, State College, PA, 2012.
    Mathematical Reviews (MathSciNet): MR3067958
  • J. Peyrière, Random beadsets and birth processes with interaction, IBM Res. Rpt. RC-7417 (1978), 1–19.
  • ––––, Process of birth interaction with the neighbors, evolution of charts, Ann. Inst. Fourier. Grenoble 31 (1981), 187–218.
    Mathematical Reviews (MathSciNet): MR644349
    Zentralblatt MATH: 0452.60089
  • ––––, Substitutions aléatoires itérées, Theor. Nombr. Bordeaux 17 (1981), 1–9.
  • ––––, Frequence des motifs dans les suites doubles invariantes par une subsitution, Ann. Math. Quebec 11 (1987), 133–138.
    Mathematical Reviews (MathSciNet): MR912167
    Zentralblatt MATH: 54.0036.03
  • M. Queffélec, Substitution dynamical systems–Spectral analysis, Lect. Notes Math. 1294, Springer-Verlag, Berlin, 1980.
    Mathematical Reviews (MathSciNet): MR2590264
  • D. Revuz, Markov chains, North-Holland, Amsterdam, 1975.
    Mathematical Reviews (MathSciNet): MR758799
    Zentralblatt MATH: 0539.60073
  • David Josiah Wing, Notions of complexity in substitution dynamical systems, Ph.D. dissertation, Oregon State University, Corvallis, 2011.
    Mathematical Reviews (MathSciNet): MR2912181
  • W. Woess, Random walks on infinite graphs and groups, Cambridge University Press, Cambridge, 2000.
    Mathematical Reviews (MathSciNet): MR1743100
    Zentralblatt MATH: 0951.60002
  • ––––, Denumerable Markov chains, European Math. Soc. Publ. House, Zürich, Switzerland, 2009.
    Mathematical Reviews (MathSciNet): MR2548569
    Zentralblatt MATH: 1219.60001

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