On a nonstandard approach to invariant measures for Markov operators

Andrzej Wiśnicki

Annales UMCS, Mathematica (2010)

  • Volume: 64, Issue: 2, page 73-80
  • ISSN: 2083-7402

Abstract

top
We show the existence of invariant measures for Markov-Feller operators defined on completely regular topological spaces which satisfy the classical positivity condition.

How to cite

top

Andrzej Wiśnicki. "On a nonstandard approach to invariant measures for Markov operators." Annales UMCS, Mathematica 64.2 (2010): 73-80. <http://eudml.org/doc/267984>.

@article{AndrzejWiśnicki2010,
abstract = {We show the existence of invariant measures for Markov-Feller operators defined on completely regular topological spaces which satisfy the classical positivity condition.},
author = {Andrzej Wiśnicki},
journal = {Annales UMCS, Mathematica},
keywords = {Markov operator; invariant measure; nonstandard analysis},
language = {eng},
number = {2},
pages = {73-80},
title = {On a nonstandard approach to invariant measures for Markov operators},
url = {http://eudml.org/doc/267984},
volume = {64},
year = {2010},
}

TY - JOUR
AU - Andrzej Wiśnicki
TI - On a nonstandard approach to invariant measures for Markov operators
JO - Annales UMCS, Mathematica
PY - 2010
VL - 64
IS - 2
SP - 73
EP - 80
AB - We show the existence of invariant measures for Markov-Feller operators defined on completely regular topological spaces which satisfy the classical positivity condition.
LA - eng
KW - Markov operator; invariant measure; nonstandard analysis
UR - http://eudml.org/doc/267984
ER -

References

top
  1. Albeverio, S., Fenstad, J. E., Høegh-Krohn, R. and Lindstrøm, T., Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, Orlando, 1986. Zbl0605.60005
  2. Anderson, R. M., Star-finite representations of measure spaces, Trans. Amer. Math. Soc. 271 (1982), 667-687. Zbl0494.28005
  3. Benci, V., Di Nasso, M. and Forti, M., The eightfold path to nonstandard analysis, Nonstandard Methods and Applications in Mathematics, Lecture Notes in Logic, 25, ASL, La Jolla, CA, 2006. Zbl1104.03061
  4. Chang, C. C., Keisler, H. J., Model Theory, 3rd edition, North-Holland, Amsterdam, 1990. 
  5. Di Nasso, M., On the foundations of nonstandard mathematics, Math. Japonica 50 (1999), 131-160. Zbl0937.03075
  6. Dudley, R. M., Real Analysis and Probability, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1989. Zbl0686.60001
  7. Foguel, S. R., Existence of invariant measures for Markov processes. II, Proc. Amer. Math. Soc. 17 (1966), 387-389. Zbl0168.16405
  8. Landers, D., Rogge, L., Universal Loeb-measurability of sets and of the standard part map with applications, Trans. Amer. Math. Soc. 304 (1987), 229-243. Zbl0633.28009
  9. Lasota, A., From fractals to stochastic differential equations, Chaos - The Interplay Between Stochastic and Deterministic Behaviour, Karpacz '95, (Eds. P. Garbaczewski, M. Wolf and A. Weron), 235-255, Springer-Verlag, Berlin, 1995. Zbl0835.60058
  10. Lasota, A., Mackey, M. C., Chaos, Fractals and Noise, Stochastic Aspects of Dynamics, Springer-Verlag, Berlin, 1994. 
  11. Lasota, A., Szarek, T., Lower bound technique in the theory of a stochastic differential equation, J. Differential Equations 231 (2006), 513-533. Zbl05115329
  12. Lasota, A., Yorke, J. A., Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam. 2 (1994), 41-77. Zbl0804.47033
  13. Loeb, P. A., Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113-122. Zbl0312.28004
  14. Loeb, P. A., Wolff, M. (Eds.), Nonstandard Analysis for the Working Mathematician, Kluwer Academic Publishers, Dordrecht, 2000. Zbl0972.03062
  15. Oxtoby, J. C., Ulam, S., On the existence of a measure invariant under a transformation, Ann. Math. 40 (1939), 560-566. Zbl0021.41202
  16. Sims, B., Ultra-techniques in Banach Space Theory, Queen's Papers in Pure and Applied Math., Vol. 60, Queen's University, Kingston, Ont., 1982. 
  17. Stettner, Ł., Remarks on ergodic conditions for Markov processes on Polish spaces, Bull. Polish Acad. Sci. Math. 42 (1994), 103-114. Zbl0815.60072
  18. Szarek, T., The stability of Markov operators on Polish spaces, Studia Math. 143 (2000), 145-152. Zbl0964.60071
  19. Szarek, T., Invariant measures for Markov operators with application to function systems, Studia Math. 154 (2003), 207-222. Zbl1036.47003
  20. Szarek, T., Invariant measures for nonexpansive Markov operators on Polish spaces, Dissertationes Math. 415 (2003), 62 pp. Zbl1051.37005

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.