Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

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• 作者：Farrukh Mukhamedov
• 关键词：Coefficient of ergodicity ; Strong ergodicity ; Weak ergodicity ; Nonhomogeneous Markov chain ; Norm ordered space
• 刊名：Positivity
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：20
• 期：1
• 页码：135-153
• 全文大小：514 KB
• 参考文献：1.Albeverio, S., Høegh-Krohn, R.: Frobenius theory for positive maps of von Neumann algebras. Commun. Math. Phys. 64, 83–94 (1978)CrossRef MATH
  2.Alfsen, E.M.: Compact Convex Sets and Booundary Integrals. Springer-Verlag, Berlin (1971)CrossRef
  3.Attal, S., Petruccione, F., Sabot, C., Sinayskiy, I.: Open quantum random walks. J. Stat. Phys. 147, 832–852 (2012)CrossRef MathSciNet MATH
  4.Ayupov, Sh, Sarymsakov, T.A.: On homogeneous Markov chains on semifields. Theor. Probab. Appl. 26, 510–520 (1982)CrossRef MATH
  5.Bartoszek, W.: Asymptotic properties of iterates of stochastic operators on (AL) Banach lattices. Anal. Polon. Math. 52, 165–173 (1990)MathSciNet MATH
  6.Bartoszek, W., Kuna, B.: Strong mixing Markov semigroups on \({\cal C}_{1}\) are meager. Colloq. Math. 105, 311–317 (2006)CrossRef MathSciNet MATH
  7.Bartoszek, W., Kuna, B.: On residualities in the set of Markov operators on \({\cal C}_{1}\) . Proc. Am. Math. Soc. 133, 2119–2129 (2005)CrossRef MathSciNet MATH
  8.Bartoszek, W., Pulka, M.: On mixing in the class of quadratic stochastic operators. Nonlin. Anal. Theor. Methods 86, 95–113 (2013)CrossRef MathSciNet MATH
  9.Berdikulov, M.: Markov processes on order-unit spaces. Theory Probab. Appl. 53, 136–144 (2009)CrossRef MathSciNet MATH
  10.Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, I. Springer, New York (1979)CrossRef
  11.Carbone, R., Sasso, E., Umanita, V.: On the asymptotic behavior of generic quantum Markov semigroups Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 17, 1450001 (2014)CrossRef MathSciNet
  12.Carbone, R., Pautrat, Y.: Open quantum random walks: reducibility, period, ergodic properties. Ann. Henri Poincare. doi:10.​1007/​s00023-015-0396-y
  13.Cohen, J.E., Iwasa, Y., Rautu, G., Ruskai, M.B., Seneta, E., Zbaganu, G.: Relative entropy under mappings by stochastic matrices. Linear Algebra Appl. 179, 211–235 (1993)CrossRef MathSciNet MATH
  14.Dobrushin, R. L.: Central limit theorem for nonstationary Markov chains. I, II. Theor. Probab. Appl. 1, 65–80, 329–383 (1956)
  15.Dorea, C.C.Y., Pereira, A.G.C.: A note on a variation of Doeblin’s condition for uniform ergodicity of Markov chains. Acta Math. Hungar. 110, 287–292 (2006)CrossRef MathSciNet MATH
  16.Emel’yanov, EYu., Wolff, M.P.H.: Positive operators on Banach spaces ordered by strongly normal cones. Positivity 7, 3–22 (2003)CrossRef MathSciNet MATH
  17.Fagnola, F., Rebolledo, R.: On the existance of stationary states for quantum dyanamical semigroups. J. Math. Phys. 42, 1296–1308 (2001)CrossRef MathSciNet MATH
  18.Fagnola, F., Rebolledo, R.: Transience and recurrence of quantum Markov semigroups, Probab. Theory Relat. Fields 126, 289–306 (2003)CrossRef MathSciNet MATH
  19.Gaubert, S.: Qu, Z.: Dobrushin’s ergodicity coefficient for Markov operators on cones and beyond, Integ. Eqs. Operator Theor. 81, 127–150 (2015)
  20.Ipsen, I.C.F., Salee, T.M.: Ergodicity coefficients defined by vector norms. SIAM J. Matrix Anal. Appl. 32, 153–200 (2011)CrossRef MathSciNet MATH
  21.Jajte, R.: Strong Linit Theorems in Non-commutative probability. Lecture Notes in Math, vol. 1110. Springer, Berlin-Heidelberg (1984)
  22.Johnson, J., Isaacson, D.: Conditions for strong ergodicity using intensity matrices. J. Appl. Probab. 25, 34–42 (1988)CrossRef MathSciNet MATH
  23.Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin-New York (1985)CrossRef MATH
  24.Łuczak, A.: Qantum dynamical semigroups in strongly finite von Neumann algebras. Acta Math. Hungar. 92, 11–17 (2001)CrossRef MathSciNet MATH
  25.Madsen, R.W., Isaacson, D.L.: Strongly ergodic behavior for non-stationary Markov processes. Ann. Probab. 1, 329–335 (1973)CrossRef MathSciNet MATH
  26.Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, Berlin (1994)
  27.Mukhamedov, F.: Dobrushin ergodicity coefficient and ergodicity of noncommutative Markov chains. J. Math. Anal. Appl. 408, 364–373 (2013)CrossRef MathSciNet MATH
  28.Mukhamedov, F.: Weak ergodicity of nonhomogeneous Markov chains on noncommutative \(L^1\) -spaces. Banach J. Math. Anal. 7, 53–73 (2013)CrossRef MathSciNet MATH
  29.Mukhamedov, F.: On \(L_1\) -weak ergodicity of nonhomogeneous discrete Markov processes and its applications. Rev. Mat. Compult. 26, 799–813 (2013)CrossRef MathSciNet
  30.Mukhamedov, F.: On \(L_1\) -weak ergodicity of nonhomogeneous continuous-time Markov processes. Bull. Iran. Math. Soc. 40, 1227–1242 (2014)MathSciNet
  31.Mukhamedov, F., Temir, S., Akin, H.: On stability properties of positive contractions of \(L^1\) -spaces accosiated with finite von Neumann algebras. Colloq. Math. 105, 259–269 (2006)CrossRef MathSciNet MATH
  32.Niculescu, C., Ströh, A., Zsidó, L.: Noncommutative extensions of classical and multiple recurrence theorems. J. Operator Theory 50, 3–52 (2003)MathSciNet MATH
  33.Nummelin, E.: General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press, Cambridge (1984)CrossRef MATH
  34.Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256, 810–864 (2009)CrossRef MathSciNet MATH
  35.Reeb, D., Kastoryano, M.J., Wolf, M.M.: Hilbert’s projective metric in quantum information theory. J. Math. Phys. 52, 082201 (2011)CrossRef MathSciNet
  36.Rhodius, A.: On ergodicity coefficients of infinite stochastic matrices. Zeit. Anal. Anwen. 19, 873–887 (2000)CrossRef MathSciNet MATH
  37.Ruskai, M.B., Szarek, S., Werner, E.: An analysis of completely positive trace-preserving maps on \(M_2\) . Linear Algebra Appl. 347, 159–187 (2002)CrossRef MathSciNet MATH
  38.Sarymsakov, T.A., Grabarnik, G.Y.: Regularity of monotonically continuous contractions acting on the von Neumann algebra. Dokl. Akad. Nauk UzSSR 5, 9–11 (1987)MathSciNet
  39.Sarymsakov, T.A., Zimakov, N.P.: Ergodic Properties of Markov Operators in Norm Ordered Spaces With a Base, In Book: Operator Algebras and Functional Spaces. Tashkent, Fan (1985)
  40.Sarymsakov, T.A., Zimakov, N.P.: Ergodic principle for the Markov semi-group in ordered normal spaces with basis. Dokl. Akad. Nauk. SSSR 289, 554–558 (1986)MathSciNet
  41.Seneta, E.: On the historical development of the theory of finite inhomogeneous Markov chains. Proc. Cambridge Philos. Soc. 74, 507–513 (1973)CrossRef MathSciNet MATH
  42.Seneta, E.: Non-negative Matrices and Markov Chains. Springer, Berlin (2006)MATH
  43.Suchanecki, Z.: An \(L^1\) extension of stochastic dynamics for irreversible systems. In: Lecture Notes in Math. vol. 1391, pp. 367–374, Springer, Berlin (1984)
  44.Szarek, T.: The stability of Markov operators on polish spaces. Studia Math. 143, 145–152 (2000)MathSciNet MATH
  45.Szehr, O., Wolf, M.M.: Perturbation bounds for quantum Markov processes and their fixed points. J. Math. Phys. 54, 032203 (2013)CrossRef MathSciNet
  46.Tan, ChP: On the weak ergodicity of nonhomogeneous Markov chains. Statis. Probab. Lett. 26, 293–295 (1996)CrossRef MATH
  47.Vardy, J.J., Watson, B.A.: Markov processes on Riesz spaces. Positivity 16, 373–391 (2012)CrossRef MathSciNet MATH
  48.Zaharopol, R., Zbaganu, G.: Dobrushin coefficients of ergodicity and asymptotically stable \(L^{1}\) -contractions. J. Theor. Probab. 99(4), 885–902 (1999)CrossRef MathSciNet
  49.Zeifman, A.I., Isaacson, D.L.: On strong ergodicity for nonhomogeneous continuous-time Markov chains. Stochast. Process. Appl. 50, 263–273 (1994)CrossRef MathSciNet MATH
  
• 作者单位：Farrukh Mukhamedov (1)
  
  1. Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box, 141, 25710, Kuantan, Pahang, Malaysia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Operator Theory
  Potential Theory
  Calculus of Variations and Optimal Control
  Econometrics
  
• 出版者：Birkh盲user Basel
• ISSN：1572-9281

文摘

In the present paper, we define an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base , and study its properties. The defined coefficient is a generalization of the well-known the Dobrushin’s ergodicity coefficient. By means of the ergodicity coefficient we provide uniform asymptotical stability conditions for nonhomogeneous discrete Markov chains (NDMC). These results are even new in case of von Neumann algebras. Moreover, we find necessary and sufficient conditions for the weak ergodicity of NDMC. Certain relations between uniform asymptotical stability and weak ergodicity are considered.