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Quasi-stationarity and quasi-ergodicity of general Markov processes

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Abstract

In this paper, we study the quasi-stationarity and quasi-ergodicity of general Markov processes. We show, among other things, that if X is a standard Markov process admitting a dual with respect to a finite measure m and if X admits a strictly positive continuous transition density p(t, x, y) (with respect to m) which is bounded in (x, y) for every t > 0, then X has a unique quasi-stationary distribution and a unique quasi-ergodic distribution. We also present several classes of Markov processes satisfying the above conditions.

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References

  1. Breyer L A, Roberts G O. A quasi-ergodic theorem for evanescent processes. Stochastic Process Appl, 1999, 84: 177–186

    Article  MATH  MathSciNet  Google Scholar 

  2. Brown A L, Page A. Elements of Functional Analysis. London-New York: Van Nostrand-Reinhold, 1970

    MATH  Google Scholar 

  3. Chen J, Jian S. Some limit theorems of killed Brownian motion. Sci China Math, 2012, 55: 497–514

    MathSciNet  Google Scholar 

  4. Chen J, Li H, Jian S. Some limit theorems for absorbing Markov processes. J Phys A Math Theor, 2012, doi: 10.1088/1751-8113/45/34/345003

    Google Scholar 

  5. Chen Z-Q, Kim P, Kumagai T. Global heat kernel estimates for symmetric jump processes. Trans Amer Math Soc, 2011, 363: 5021–5055

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen Z-Q, Kumagai T. A priori Hölder estimate, parabolic Hanack principle and heat kernel estimates for diffusions with jumps. Rev Mat Iberoamericana, 2010, 26: 551–589

    Article  MATH  MathSciNet  Google Scholar 

  7. Darroch J N, Seneta E. On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J Appl Probab, 1965, 2: 88–100

    Article  MATH  MathSciNet  Google Scholar 

  8. Darroch J N, Seneta E. On quasi-stationary distributions in absorbing continous-time finite Markov chains. J Appl Probab, 1967, 4: 192–196

    Article  MATH  MathSciNet  Google Scholar 

  9. Davies E B. Heat Kernels and Spectral Theory. Cambridge: Cambridge University Press, 1989

    MATH  Google Scholar 

  10. Flaspohler D C. Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann Inst Statist Math, 1973, 26: 351–356

    Article  MathSciNet  Google Scholar 

  11. Kim P, Song R. Two-sided estimates on the density of Brownian motion with singular drift. Illinois J Math, 2006, 50: 635–688

    MATH  MathSciNet  Google Scholar 

  12. Kim P, Song R. On dual processes of non-symmetric diffusions with measure-valued drifts. Stochastic Process Appl, 2008, 118: 790–817

    Article  MATH  MathSciNet  Google Scholar 

  13. Kim P, Song R. Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains. Tohoku Math J, 2008, 60: 527–547

    Article  MATH  MathSciNet  Google Scholar 

  14. Kim P, Song R. Intrinsic ultracontractivity of non-symmetric diffusions with measure-valued drifts and potentials. Ann Probab, 2008, 36: 1904–1945

    Article  MATH  MathSciNet  Google Scholar 

  15. Kim P, Song R. Intrinsic ultracontractivity for non-symmetric Lévy processes. Forum Math, 2009, 21: 43–66

    MATH  MathSciNet  Google Scholar 

  16. Kim P, Song R. Stable process with singular drift. Stochastic Process Appl, 2014, 124: 2479–2516

    Article  MathSciNet  Google Scholar 

  17. Kim P, Song R. Dirichlet heat kernel estimates for stable processes with singular drift in unbounded C 1,1 open sets. Potential Anal, 2014, in press, doi: 10.1007/s11118-013-9393-4

    Google Scholar 

  18. Knobloch R, Partzsch L. Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Anal, 2010, 33: 107–136

    MATH  MathSciNet  Google Scholar 

  19. Ma Z-M, Röckner M. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Berlin: Springer, 1991

    Google Scholar 

  20. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer, 1983

    MATH  Google Scholar 

  21. Pinsky R G. The lifetimes of conditioned diffusion processes. Ann Inst H Poincaré Probab Statist, 1990, 26: 87–99

    MATH  MathSciNet  Google Scholar 

  22. Schaefer H H. Banach Lattices and Positive Operators. New York: Springer, 1974

    MATH  Google Scholar 

  23. Seneta E, Vere-Jones D. On quasi-stationary distributions in discrete-time Markov chains with a denumberable infinity of states. J Appl Probab, 1966, 3: 403–434

    MATH  MathSciNet  Google Scholar 

  24. van Doorn E A, Pollett P K. Quasi-stationary distributions. http://doc.utwente.nl/77521, 2011

    Google Scholar 

  25. Vondraeck Z. Basic potential theory of certain non-symmetric strictly α-stable processes. Glas Mat, 2002, 37: 211–233

    Google Scholar 

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Authors and Affiliations

  1. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

    JunFei Zhang, ShouMei Li & RenMing Song

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    JunFei Zhang & RenMing Song

Authors
  1. JunFei Zhang
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  2. ShouMei Li
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  3. RenMing Song
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Correspondence to ShouMei Li.

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Zhang, J., Li, S. & Song, R. Quasi-stationarity and quasi-ergodicity of general Markov processes. Sci. China Math. 57, 2013–2024 (2014). https://doi.org/10.1007/s11425-014-4835-x

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  • DOI: https://doi.org/10.1007/s11425-014-4835-x

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