Advertisement

Invariant, Super and Quasi-martingale Functions of a Markov Process

  • Lucian Beznea
  • Iulian CîmpeanEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. Finally, using the co-excessive functions, we present a two-step approach to the existence of invariant probability measures.

Keywords

Semimartingale Quasimartingale Markov process Invariant function Invariant measure 

Mathematics Subject Classification (2010)

60J45 31C05 60J40 60J25 37C40 37L40 31C25 

Notes

Acknowledgements

The first named author acknowledges support from the Romanian National Authority for Scientific Research, project number PN-III-P4-ID-PCE-2016-0372. The second named author acknowledges support from the Romanian National Authority for Scientific Research, project number PN-II-RU-TE-2014-4-0657.

References

  1. 1.
    Doob, J.L.: Classical potential theory and its probabilistic counterpart. Springer, Berlin (2001)Google Scholar
  2. 2.
    Le Gall, J.F.: Intégration, probabilités et processus aléatoires. Ecole Normale Supérieure de Paris. Sept 2006Google Scholar
  3. 3.
    Çinlar, E., Jacod, J., Protter, P., Sharpe, M.J.: Semimartingales and Markov processes. Z. Wahrscheinlichkietstheorie verw. Gebiete 54, 161–219 (1980)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Da Prato, G., Zabczyk, J.: Ergodicity for infinite dimensional systems. Cambridge University Press, Cambridge (1996)Google Scholar
  5. 5.
    Hairer, M.: Convergence of Markov processes. Lecture Notes, University of Warwick (2010). http://www.hairer.org/notes/Convergence.pdf
  6. 6.
    Komorowski, T., Peszat, S., Szarek, T.: On ergodicity of some Markov processes. Ann. Probab. 38, 1401–1443 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lasota, A., Szarek, T.: Lower bound technique in the theory of a stochastic differential equation. J. Differ. Equ. 231, 513–533 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Meyn, S.P., Tweedie, R.L.: Markov chains and stochastic stability. Springer, London (1993)CrossRefGoogle Scholar
  9. 9.
    Meyn, S.P., Tweedie, R.L.: Stability of markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Probab. 25, 487–517 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Meyn, S.P., Tweedie, R.L.: Stability of markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25, 518–548 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Beznea, L., Cîmpean, I., Röckner, M.: A new approach to the existence of invariant measures for Markovian semigroups. arXiv:1508.06863v3
  12. 12.
    Beznea, L., Cîmpean, I.: Quasimartingales associated to Markov processes. Trans. Am. Math. Soc. (to appear (2017)). arXiv:1702.06282
  13. 13.
    Beznea, L., Cîmpean, I., Röckner, M.L: Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents. Stoch. Process. Appl. (2017). arXiv:1409.6492v2.  https://doi.org/10.1016/j.spa.2017.07.009
  14. 14.
    Protter, P.E.: Stochastic integration and differential equations. Springer, Berlin (2005)Google Scholar
  15. 15.
    Blumenthal, R., Getoor, R.: Markov processes and potential theory. Academic Press, New York (1968)Google Scholar
  16. 16.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. Walter de Gruyter, Berlin/New York (2011)zbMATHGoogle Scholar
  17. 17.
    Ma, Z.M., Overbeck, L., Röckner, M.: Markov processes associated with semi-Dirichlet forms. Osaka J. Math. 32, 97–119 (1995)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ma, Z.M., Röckner, M.: An introduction to the theory of (non-symmetric) Dirichlet forms. Springer, Berlin (1992)CrossRefGoogle Scholar
  19. 19.
    Oshima, Y.: Semi-dirichlet forms and Markov processes. Walter de Gruyter and Co., Berlin (2013)CrossRefGoogle Scholar
  20. 20.
    Beznea, L., Boboc, N.: Potential theory and right processes. Springer Series, Mathematics and its applications (572). Kluwer, Dordrecht (2004)Google Scholar
  21. 21.
    Beznea, L., Boboc, N., Röckner, M.: Quasi-regular Dirichlet forms and \(L^p\)-resolvents on measurable spaces. Potential Anal. 25, 269–282 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fukushima, M.: On semi-martingale characterization of functionals of symmetric Markov processes. Electron. J. Probab. 4, 1–32 (1999)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bass, R.F., Hsu, P.: The semimartingale structure of reflecting Brownian motion. Proc. Am. Math. Soc. 108, 1007–1010 (1990)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chen, Z.Q.: On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Relat. Fields 94, 281–315 (1993)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Chen, Z.Q., Fitzsimmons, P.J., Williams, R.J.: Reflecting Brownian motions: quasimartingales and strong Caccioppoli sets. Potential Anal. 2, 219–243 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pardoux, E., Williams, R.J.: Symmetric reflected diffusions. Ann. Inst. H. Poincaré Probab. Stat. 30, 13–62 (1994)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Williams, R.J., Zheng, W.A.: On reflecting Brownian motion - a weak convergence approach. Ann. Inst. Henri Poincaré 26, 461–488 (1990)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Trutnau, G.: Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part. Probab. Theory Relat. Fields 127, 455–495 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Fukushima, M.: BV functions and distorted Ornstein Uhlenbeck processes over the abstract Wiener space. J. Funct. Anal. 174, 227–249 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Fukushima, M., Hino, M.: On the space of BV functions and a related stochastic calculus in infinite dimensions. J. Funct. Anal. 183, 245–268 (2001)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Röckner, M., Zhu, R., Zhu, X.: The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple. Ann. Probab. 40, 1759–1794 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Röckner, M., Zhu, R., Zhu, X.: BV functions in a Gelfand triple for differentiable measure and its applications. Forum Math. 27, 1657–1687 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Albeverio, S., Kondratiev, Y.G., Röckner, M.: Ergodicity of \(L^2\)-semigroups and extremality of Gibbs states. J. Funct. Anal. 144, 394–423 (1997)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Schilling, R.L.: A note on invariant sets. Probab. Math. Statist. 24, 47–66 (2004)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \(L^p\)-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Beznea, L., Röckner, M.: On the existence of the dual right Markov process and applications. Potential Anal. 42, 617–627 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Röckner, M., Trutnau, G.: A remark on the generator of a right-continuous Markov process. Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 10, 633–640 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Komlós, J.: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18, 217–229 (1967)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Simion Stoilow Institute of Mathematics of the Romanian AcademyBucharestRomania
  2. 2.University of Bucharest, Faculty of Mathematics and Computer Science, and Centre Francophone en Mathématique de BucarestBucharestRomania

Personalised recommendations