# American Institute of Mathematical Sciences

September  2016, 21(7): 2337-2361. doi: 10.3934/dcdsb.2016050

## Stochastic dynamics: Markov chains and random transformations

 1 Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, United States, United States, United States

Received  June 2015 Revised  February 2016 Published  August 2016

This article outlines an attempt to lay the groundwork for understanding stochastic dynamical descriptions of biological processes in terms of a discrete-state space, discrete-time random dynamical system (RDS), or random transformation approach. Such mathematics is not new for continuous systems, but the discrete state space formulation significantly reduces the technical requirements for its introduction to a much broader audiences. In particular, we establish some elementary contradistinctions between Markov chain (MC) and RDS descriptions of a stochastic dynamics. It is shown that a given MC is compatible with many possible RDS, and we study in particular the corresponding RDS with maximum metric entropy. Specifically, we show an emergent behavior of an MC with a unique absorbing and aperiodic communicating class, after all the trajectories of the RDS synchronizes. In biological modeling, it is now widely acknowledged that stochastic dynamics is a more complete description of biological reality than deterministic equations; here we further suggest that the RDS description could be a more refined description of stochastic dynamics than a Markov process. Possible applications of discrete-state RDS are systems with fluctuating law of motion, or environment, rather than inherent stochastic movements of individuals.
Citation: Felix X.-F. Ye, Yue Wang, Hong Qian. Stochastic dynamics: Markov chains and random transformations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2337-2361. doi: 10.3934/dcdsb.2016050
##### References:
 [1] L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, $2^{nd}$ edition, CRC Press, Boca Raton, FL, 2011.  Google Scholar [2] L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [3] L. Arnold and H. Crauel, Random dynamical systems, in Lyapunov Exponents, (eds. L. Arnold, H. Crauel and J.-P. Eckmann), Springer, Berlin, 1486 (2006), 1-22. doi: 10.1007/BFb0086654.  Google Scholar [4] P. H. Baxendale, Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms. Spatial stochastic processes, Progr. Probab., 19 (1991), 189-218.  Google Scholar [5] R. Bhattacharya and M. Majumdar, Random Dynamical Systems: Theory and Applications, Cambridge Univ. Press, U.K., 2007. doi: 10.1017/CBO9780511618628.  Google Scholar [6] G. Birkhoff, Three observations on linear algebra, Univ. Nac. Tucumán. Revista A, 5 (1946), 147-151.  Google Scholar [7] R. M Blumenthal and H. K. 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Gallavotti, Statistical Mechanics: A Short Treatise, Springer, New York, 1999. doi: 10.1007/978-3-662-03952-6.  Google Scholar [14] H. Ge, M. Qian and H. Qian, Stochastic theory of nonequilibrium steady states (Part II): Applications in chemical biophysics, Phys. Rep., 510 (2012), 87-118. doi: 10.1016/j.physrep.2011.09.001.  Google Scholar [15] B. Hasselblatt and A. Katok, Principal structures, in Handbook of Dynamical Systems, vol. 1A (eds. B. Hasselblatt and A. Katok), Elsevier, Amsterdam, 1 (2002), 1-203. doi: 10.1016/S1874-575X(02)80003-0.  Google Scholar [16] W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer, New York, 1984.  Google Scholar [17] E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover, New York, 1966.  Google Scholar [18] S. L. Kalpazidou, Cycle Representations of Markov Processes, $2^{nd}$ edition, Springer, New York, 2006.  Google Scholar [19] M. Keane, Ergodic theory and subshifts of finite type, in Ergodic theory, Symbolic Dynamics and Hyperbolic Spaces (eds. T. Bedford, M. Keane and C. Series), Oxford, (1991), 35-70.  Google Scholar [20] A. I. Khinchin, Mathematical Foundations of Information Theory, Dover, New York, 1957.  Google Scholar [21] Yu. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, Basel, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar [22] Yu. Kifer and P.-D. Liu, Random dynamics, in Handbook of Dynamical Systems, (eds. B. Hasselblatt and A. Katok), Elsevier, Amsterdam, 1 (2006), 379-499. doi: 10.1016/S1874-575X(06)80030-5.  Google Scholar [23] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar [24] V. Lecomte, C. Appert-Rolland and F. van Wijland, Thermodynamic formalism for systems with Markov dynamics, J. Stat. Phys., 127 (2007), 51-106. doi: 10.1007/s10955-006-9254-0.  Google Scholar [25] T. Liggett, The coupling technique in interacting particle systems, In Doeblin and Modern Probability, (ed. H. Cohn), AMS, Providence, 149 (1993), 73-83. doi: 10.1090/conm/149/01271.  Google Scholar [26] K. K. Lin, Stimulus-response reliability of biological networks, in Nonautonomous and Random Dynamical Systems in Life Sciences (eds. P. Kloeden and C. Poetzsche), Springer, New York, 2102 (2012), 135-161. doi: 10.1007/978-3-319-03080-7_4.  Google Scholar [27] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, U.K., 1995. doi: 10.1017/CBO9780511626302.  Google Scholar [28] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, UK, 2006. doi: 10.1142/p473.  Google Scholar [29] M. Marcus, H. Minc and B. Moyls, Some results on non-negative matrices, J. Res. Nat. Bur. Standards Sec. B, 65 (1961), 205-209. doi: 10.6028/jres.065B.019.  Google Scholar [30] M. L. Mehta, Random Matrices, $3^{rd}$ edition, Academic Press, New York, 2004.  Google Scholar [31] H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.  Google Scholar [32] J. A. Morrison and J. McKenna, Analysis of some stochastic ordinary differential equations, in Stochastic Differential Equations, SIAM-AMS Proc., Vol. 6 (eds. J. B. Keller and H. P. McKean), Amer. Math. Soc., Providence, R.I., (1973), 97-161.  Google Scholar [33] J. D. Murray, Mathematical Biology: I. An Introduction, $3^{rd}$ edition, Springer, New York, 2002.  Google Scholar [34] D. S. Ornstein, Ergodic theory, randomness, and "chaos'', Science, 243 (1989), 182-187. doi: 10.1126/science.243.4888.182.  Google Scholar [35] Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, in Lyapunov Exponents, (eds. L. Arnold, H. Crauel and J.-P. Eckmann), Springer, Berlin, 1486 (1991), 64-80. doi: 10.1007/BFb0086658.  Google Scholar [36] M. A. Pinsky, Lectures on Random Evolution, World Scientific, Singapore, 1991. doi: 10.1142/1328.  Google Scholar [37] H. Qian and J. A. Schellman, Helix-coil theories: A comparative studies for finite length polypeptides, J. Phys. Chem., 96 (1992), 3987-3994. doi: 10.1021/j100189a015.  Google Scholar [38] H. Qian, The mathematical theory of molecular motor movement and chemomechanical energy transduction, J. Math. Chem., 27 (2000), 219-234. doi: 10.1023/A:1026428320489.  Google Scholar [39] H. Qian, Cooperativity in cellular biochemical processes: Noise-enhanced sensitivity, fluctuating enzyme, bistability with nonlinear feedback, and other mechanisms for sigmoidal responses, Annu. Rev. Biophys., 41 (2012), 179-204. doi: 10.1146/annurev-biophys-050511-102240.  Google Scholar [40] H. Qian, S. Kjelstrup, A. B. Kolomeisky and D. Bedeaux, Entropy production in mesoscopic stochastic thermodynamics | Nonequilibrium steady state cycles driven by chemical potentials, temperatures, and mechanical forces, J. Phys. Cond. Matt. 28 (2016), 153004. arXiv:1601.04018 Google Scholar [41] M. Qian, J.-S. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms, Lec. Notes Math. vol. 1978, Springer, New York, 2009. doi: 10.1007/978-3-642-01954-8.  Google Scholar [42] M. Qian and F.-X. Zhang, Entropy production rate of the coupled diffusion process, J. Theor. Probab., 24 (2011), 729-745. doi: 10.1007/s10959-011-0352-9.  Google Scholar [43] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, NJ, 1970.  Google Scholar [44] M. Santillán and H. Qian, Irreversible thermodynamics in multiscale stochastic dynamical systems, Phys. Rev. E, 83 (2011), 041130. arXiv:1003.3513  Google Scholar [45] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, Boulder, 2001. Google Scholar [46] A. Swishchuk and S. Islam, Random Dynamical Systems in Finance, Chapman & Hall/CRC, New York, 2013. doi: 10.1201/b14989.  Google Scholar [47] H. Thorisson, Coupling and shift-coupling random sequences, Doeblin and Modern Probability, (ed. H. Cohn), AMS, Providence, 149 (1993), 85-95. doi: 10.1090/conm/149/01280.  Google Scholar [48] J. M. van Campenhout and T. M. Cover, Maximum entropy and conditional probability, IEEE Infor. Th., IT-27 (1981), 483-489. doi: 10.1109/TIT.1981.1056374.  Google Scholar [49] J. van Neumann, The general and logical theory of automata, Cerebral Mechanisms in Behavior, The Hixon Symposium, pp. 1-31; discussion, pp. 32-41. John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951.  Google Scholar [50] P. Walters, An Introduction to Ergodic Theory, Spinger, New York, 1982.  Google Scholar [51] S. Wolfram, A New Kind of Science, Wolfram media, Champaign, 2002.  Google Scholar [52] G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.  Google Scholar [53] X.-J. Zhang, H. Qian and M. Qian, Stochastic theory of nonequilibrium steady states and its applications (Part I), Phys. Rep., 510 (2012), 1-86. Google Scholar [54] X.-J. Zhang, M. Qian and H. Qian, Stochastic dynamics of electrical membrane with voltage-dependent ion channel fluctuations, Europhys. Lett., 106 (2014), 10002. arXiv:1404.1548 Google Scholar

show all references

##### References:
 [1] L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, $2^{nd}$ edition, CRC Press, Boca Raton, FL, 2011.  Google Scholar [2] L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [3] L. Arnold and H. Crauel, Random dynamical systems, in Lyapunov Exponents, (eds. L. Arnold, H. Crauel and J.-P. Eckmann), Springer, Berlin, 1486 (2006), 1-22. doi: 10.1007/BFb0086654.  Google Scholar [4] P. H. Baxendale, Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms. Spatial stochastic processes, Progr. Probab., 19 (1991), 189-218.  Google Scholar [5] R. Bhattacharya and M. Majumdar, Random Dynamical Systems: Theory and Applications, Cambridge Univ. Press, U.K., 2007. doi: 10.1017/CBO9780511618628.  Google Scholar [6] G. Birkhoff, Three observations on linear algebra, Univ. Nac. Tucumán. Revista A, 5 (1946), 147-151.  Google Scholar [7] R. M Blumenthal and H. K. Corson, On continuous collections of measures, Proc. 6th Berkeley Symp. on Math. Stat. and Prob., 2 (1972), 33-40.  Google Scholar [8] K.-S. Chan and H. Tong, Chaos: A Statistical Perspective, Springer, New York, 2001. doi: 10.1007/978-1-4757-3464-5.  Google Scholar [9] Y.-D. Chen, Asymmetry and external noise-induced free energy transduction, Proc. Natl. Acad. Sci. U.S.A., 84 (1987), 729-733. doi: 10.1073/pnas.84.3.729.  Google Scholar [10] T. Downarowicz, Entropy in Dynamical Systems, Cambridge Univ. Press, UK, 2011. doi: 10.1017/CBO9780511976155.  Google Scholar [11] S. P. Ellner and J. Guckenheimer, Dynamic Models in Biology, Princeton Univ. Press, NJ, 2006.  Google Scholar [12] G. Froyland, Extracting dynamical behavior via Markov models, Nonlinear dynamics and statistics (Cambridge, 1998), Birkhäuser Boston, Boston, MA, (2001), 281-321.  Google Scholar [13] G. Gallavotti, Statistical Mechanics: A Short Treatise, Springer, New York, 1999. doi: 10.1007/978-3-662-03952-6.  Google Scholar [14] H. Ge, M. Qian and H. Qian, Stochastic theory of nonequilibrium steady states (Part II): Applications in chemical biophysics, Phys. Rep., 510 (2012), 87-118. doi: 10.1016/j.physrep.2011.09.001.  Google Scholar [15] B. Hasselblatt and A. Katok, Principal structures, in Handbook of Dynamical Systems, vol. 1A (eds. B. Hasselblatt and A. Katok), Elsevier, Amsterdam, 1 (2002), 1-203. doi: 10.1016/S1874-575X(02)80003-0.  Google Scholar [16] W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology, Springer, New York, 1984.  Google Scholar [17] E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover, New York, 1966.  Google Scholar [18] S. L. Kalpazidou, Cycle Representations of Markov Processes, $2^{nd}$ edition, Springer, New York, 2006.  Google Scholar [19] M. Keane, Ergodic theory and subshifts of finite type, in Ergodic theory, Symbolic Dynamics and Hyperbolic Spaces (eds. T. Bedford, M. Keane and C. Series), Oxford, (1991), 35-70.  Google Scholar [20] A. I. Khinchin, Mathematical Foundations of Information Theory, Dover, New York, 1957.  Google Scholar [21] Yu. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, Basel, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar [22] Yu. Kifer and P.-D. Liu, Random dynamics, in Handbook of Dynamical Systems, (eds. B. Hasselblatt and A. Katok), Elsevier, Amsterdam, 1 (2006), 379-499. doi: 10.1016/S1874-575X(06)80030-5.  Google Scholar [23] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, $2^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar [24] V. Lecomte, C. Appert-Rolland and F. van Wijland, Thermodynamic formalism for systems with Markov dynamics, J. Stat. Phys., 127 (2007), 51-106. doi: 10.1007/s10955-006-9254-0.  Google Scholar [25] T. Liggett, The coupling technique in interacting particle systems, In Doeblin and Modern Probability, (ed. H. Cohn), AMS, Providence, 149 (1993), 73-83. doi: 10.1090/conm/149/01271.  Google Scholar [26] K. K. Lin, Stimulus-response reliability of biological networks, in Nonautonomous and Random Dynamical Systems in Life Sciences (eds. P. Kloeden and C. Poetzsche), Springer, New York, 2102 (2012), 135-161. doi: 10.1007/978-3-319-03080-7_4.  Google Scholar [27] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, U.K., 1995. doi: 10.1017/CBO9780511626302.  Google Scholar [28] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, UK, 2006. doi: 10.1142/p473.  Google Scholar [29] M. Marcus, H. Minc and B. Moyls, Some results on non-negative matrices, J. Res. Nat. Bur. Standards Sec. B, 65 (1961), 205-209. doi: 10.6028/jres.065B.019.  Google Scholar [30] M. L. Mehta, Random Matrices, $3^{rd}$ edition, Academic Press, New York, 2004.  Google Scholar [31] H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.  Google Scholar [32] J. A. Morrison and J. McKenna, Analysis of some stochastic ordinary differential equations, in Stochastic Differential Equations, SIAM-AMS Proc., Vol. 6 (eds. J. B. Keller and H. P. McKean), Amer. Math. Soc., Providence, R.I., (1973), 97-161.  Google Scholar [33] J. D. Murray, Mathematical Biology: I. An Introduction, $3^{rd}$ edition, Springer, New York, 2002.  Google Scholar [34] D. S. Ornstein, Ergodic theory, randomness, and "chaos'', Science, 243 (1989), 182-187. doi: 10.1126/science.243.4888.182.  Google Scholar [35] Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, in Lyapunov Exponents, (eds. L. Arnold, H. Crauel and J.-P. Eckmann), Springer, Berlin, 1486 (1991), 64-80. doi: 10.1007/BFb0086658.  Google Scholar [36] M. A. Pinsky, Lectures on Random Evolution, World Scientific, Singapore, 1991. doi: 10.1142/1328.  Google Scholar [37] H. Qian and J. A. Schellman, Helix-coil theories: A comparative studies for finite length polypeptides, J. Phys. Chem., 96 (1992), 3987-3994. doi: 10.1021/j100189a015.  Google Scholar [38] H. Qian, The mathematical theory of molecular motor movement and chemomechanical energy transduction, J. Math. Chem., 27 (2000), 219-234. doi: 10.1023/A:1026428320489.  Google Scholar [39] H. Qian, Cooperativity in cellular biochemical processes: Noise-enhanced sensitivity, fluctuating enzyme, bistability with nonlinear feedback, and other mechanisms for sigmoidal responses, Annu. Rev. Biophys., 41 (2012), 179-204. doi: 10.1146/annurev-biophys-050511-102240.  Google Scholar [40] H. Qian, S. Kjelstrup, A. B. Kolomeisky and D. Bedeaux, Entropy production in mesoscopic stochastic thermodynamics | Nonequilibrium steady state cycles driven by chemical potentials, temperatures, and mechanical forces, J. Phys. Cond. Matt. 28 (2016), 153004. arXiv:1601.04018 Google Scholar [41] M. Qian, J.-S. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms, Lec. Notes Math. vol. 1978, Springer, New York, 2009. doi: 10.1007/978-3-642-01954-8.  Google Scholar [42] M. Qian and F.-X. Zhang, Entropy production rate of the coupled diffusion process, J. Theor. Probab., 24 (2011), 729-745. doi: 10.1007/s10959-011-0352-9.  Google Scholar [43] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, NJ, 1970.  Google Scholar [44] M. Santillán and H. Qian, Irreversible thermodynamics in multiscale stochastic dynamical systems, Phys. Rev. E, 83 (2011), 041130. arXiv:1003.3513  Google Scholar [45] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, Boulder, 2001. Google Scholar [46] A. Swishchuk and S. Islam, Random Dynamical Systems in Finance, Chapman & Hall/CRC, New York, 2013. doi: 10.1201/b14989.  Google Scholar [47] H. Thorisson, Coupling and shift-coupling random sequences, Doeblin and Modern Probability, (ed. H. Cohn), AMS, Providence, 149 (1993), 85-95. doi: 10.1090/conm/149/01280.  Google Scholar [48] J. M. van Campenhout and T. M. Cover, Maximum entropy and conditional probability, IEEE Infor. Th., IT-27 (1981), 483-489. doi: 10.1109/TIT.1981.1056374.  Google Scholar [49] J. van Neumann, The general and logical theory of automata, Cerebral Mechanisms in Behavior, The Hixon Symposium, pp. 1-31; discussion, pp. 32-41. John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951.  Google Scholar [50] P. Walters, An Introduction to Ergodic Theory, Spinger, New York, 1982.  Google Scholar [51] S. Wolfram, A New Kind of Science, Wolfram media, Champaign, 2002.  Google Scholar [52] G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.  Google Scholar [53] X.-J. Zhang, H. Qian and M. Qian, Stochastic theory of nonequilibrium steady states and its applications (Part I), Phys. Rep., 510 (2012), 1-86. Google Scholar [54] X.-J. Zhang, M. Qian and H. Qian, Stochastic dynamics of electrical membrane with voltage-dependent ion channel fluctuations, Europhys. Lett., 106 (2014), 10002. arXiv:1404.1548 Google Scholar
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