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Realization of an Ergodic Markov Chain as a Random Walk Subject to a Synchronizing Road Coloring

Part of: Markov processes Graph theory Ergodic theory Random dynamical systems

Published online by Cambridge University Press:  14 July 2016

Kouji Yano  and
Kenji Yasutomi
Kouji Yano*
Affiliation:
Kobe University
Kenji Yasutomi*
Affiliation:
Ritsumeikan University
*
∗ Current address: Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji Higashi, Kusatsu, Shiga 525-8577, Japan.
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Abstract

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An ergodic Markov chain is proved to be the realization of a random walk in a directed graph subject to a synchronizing road coloring. The result ensures the existence of appropriate random mappings in Propp-Wilson's coupling from the past. The proof is based on the road coloring theorem. A necessary and sufficient condition for approximate preservation of entropies is also given.

Keywords

Markov chainrandom walk in a directed graphroad coloring problemTsirelson's equationcoupling from the past

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 37A35: Entropy and other invariants, isomorphism, classification 05C81: Random walks on graphs 37H10: Generation, random and stochastic difference and differential equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 766 - 777
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Research supported by KAKENHI (20740060).

References

[1] Adler, R. L., Goodwyn, L. W. and Weiss, B. (1977). Equivalence of topological Markov shifts. Israel J. Math., 27, 48–63.Google Scholar
[2] Adler, R. L. and Weiss, B. (1970). Similarity of Automorphisms of the Torus (Memoirs Amer. Math. Soc. 98). American Mathematical Society, Providence, RI.Google Scholar
[3] Akahori, J., Uenishi, C. and Yano, K. (2008). Stochastic equations on compact groups in discrete negative time. Prob. Theory Relat. Fields 140, 569–593.Google Scholar
[4] Billingsley, P. (1978). Ergodic Theory and Information. Robert E. Krieger Publishing, Huntington, NY.Google Scholar
[5] Budzban, G. (2004). Semigroups and the generalized road coloring problem. Semigroup Forum 69, 201–208.Google Scholar
[6] Budzban, G. and Feinsilver, P. (2007). Completely simple semigroups, Lie algebras, and the road coloring problem. Semigroup Forum 74, 206–226.Google Scholar
[7] Budzban, G. and Mukherjea, A. (2000). A semigroup approach to the road coloring problem. In Probability on Algebraic Structures (Gainesville, FL, 1999; Contemp. Math. 261), American Mathematical Society, Providence, RI, pp. 195–207.Google Scholar
[8] Culik, K. II, Karhumäki, J. and Kari, J. (2002). A note on synchronized automata and road coloring problem. In Developments in Language Theory (Vienna, 2001; Lecture Notes Comput. Sci. 2295), Springer, Berlin, pp. 175–185.Google Scholar
[9] Friedman, J. (1990). On the road coloring problem. Proc. Amer. Math. Soc. 110, 1133–1135.Google Scholar
[10] Friedman, N. A. and Ornstein, D. S. (1970). On isomorphism of weak Bernoulli transformations. Adv. Math. 5, 365–394.Google Scholar
[11] Häggström, O. (2002). Finite Markov Chains and Algorithmic Applications (London Math. Soc. Student Texts 52). Cambridge University Press.Google Scholar
[12] Hirayama, T. and Yano, K. (2010). Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups. Stoch. Process. Appl. 120, 1404–1423.Google Scholar
[13] Hirayama, T. and Yano, K. (2010). Strong solutions of Tsirelson's equation in discrete time taking values in compact spaces with semigroup action. Preprint. Available at http://arxiv.org/abs/1005.0038v1.Google Scholar
[14] Ornstein, D. (1970). Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4, 337–352.Google Scholar
[15] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Internat. Random Structures Algorithms 9, 223–252.Google Scholar
[16] Rosenblatt, M. (1959). Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8, 665–681.Google Scholar
[17] Rosenblatt, M. (1962). Stationary Markov chains and independent random variables. J. Math. Mech. 9, 945–949. (Addendum: 11 (1962), 317.)Google Scholar
[18] Trahtman, A. N. (2009). The road coloring problem. Israel J. Math. 172, 51–60.Google Scholar
[19] Yano, K. (2010). Random walk in a finite directed graph subject to a road coloring. Preprint. Available at http://arxiv.org/abs/1005.0079v3.Google Scholar
[20] Yano, K. and Takahashi, Y. (2007). Time evolution with and without remote past. Sūrikaisekikenkyūsho Kōkyūroku 1552, 164–171.Google Scholar
[21] Yano, K. and Yor, M. (2010). Around Tsirelson's equation, or: the evolution process may not explain everything. Preprint. Available at http://arxiv.org/abs/0906.3442v2.Google Scholar
[22] Yor, M. (1992). Tsireĺson's equation in discrete time. Probab. Theory Relat. Fields 91, 135–152.Google Scholar