Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique

Jean-Pierre Conze; Yves Guivarc'h

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 2, page 457-480
  • ISSN: 0010-1354

Abstract

top
The invariant measures for a Markovian operator corresponding to a random walk, in a random stationary one-dimensional environment defined by a dynamical system, are quasi-invariant measures for the system. We discuss the construction of such measures in the general case and show unicity, under some assumptions, for a rotation on the circle.

How to cite

top

Conze, Jean-Pierre, and Guivarc'h, Yves. "Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique." Colloquium Mathematicae 84/85.2 (2000): 457-480. <http://eudml.org/doc/210826>.

@article{Conze2000,
author = {Conze, Jean-Pierre, Guivarc'h, Yves},
journal = {Colloquium Mathematicae},
keywords = {random walk; Markovian operator; quasi-invariant measures},
language = {fre},
number = {2},
pages = {457-480},
title = {Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique},
url = {http://eudml.org/doc/210826},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Conze, Jean-Pierre
AU - Guivarc'h, Yves
TI - Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 2
SP - 457
EP - 480
LA - fre
KW - random walk; Markovian operator; quasi-invariant measures
UR - http://eudml.org/doc/210826
ER -

References

top
  1. [1] S. Alili, Processus de branchement et marche aléatoire en milieux désordonnés, thèse, Université Pierre et Marie Curie (Paris VI), 1993. 
  2. [2] D. Anosov, On the additive functional homological equation associated with an irrational rotation of the circle, Izv. Akad. Nauk SSSR 37 (1973), 1259-1274 (in Russian). Zbl0298.28016
  3. [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975. Zbl0308.28010
  4. [4] J. Brémont, Comportement des sommes ergodiques pour des rotations et des fonctions continues peu régulières, Publications des Séminaires de Rennes, 1999. 
  5. [5] J.-P. Conze, Equirépartition et ergodicité de transformations cylindriques, Publications des Séminaires de Rennes, 1976. 
  6. [6] J.-P. Conze et Y. Guivarc'h, Croissance des sommes ergodiques et principe va- riationnel, preprint, Rennes, 1997. 
  7. [7] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math. 527, Springer, 1976. Zbl0328.28008
  8. [8] H. Federer, Geometric Measure Theory, Classics in Math., Springer, 1996. Zbl0874.49001
  9. [9] Y. Guivarc'h et J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincaré Sér. Probab. Statist. 24 (1988), 73-98. Zbl0649.60041
  10. [10] G. Halász, Remarks on the remainder in Birkhoff's ergodic theorem, Acta Math. Acad. Sci. Hungar. 28 (1976), 389-395. Zbl0336.28005
  11. [11] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. Inst. Hautes Etudes Sci. 49 (1979), 5-233. Zbl0448.58019
  12. [12] H. Kesten, M. V. Kozlov and F. Spitzer, A limit law for random walks in a random environment, Compositio Math. 30 (1975), 145-168. Zbl0388.60069
  13. [13] S. M. Kozlov, The method of averaging and walks in inhomogeneous environments, Russian Math. Surveys 40 (1985), no. 2, 73-145. Zbl0615.60063
  14. [14] S. M. Kozlov and S. A. Molchanov, On conditions for applicability of the central limit theorem to random walks on a lattice, Soviet Math. Dokl. 30 (1984), 410-413. Zbl0603.60020
  15. [15] M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff, Gro- ningen, 1964. 
  16. [16] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. 
  17. [17] A. V. Letchikov, A criterion for applicability of the CLT to one-dimensional random walks in random environments, Theory Probab. Appl. 37 (1992), 553-557. Zbl0787.60088
  18. [18] W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. 25, Springer, 1993. Zbl0791.58003
  19. [19] S. A. Molchanov, Lectures on random media, in: Lectures on Probability Theory (Saint-Flour, 1992), Lecture Notes in Math. 1581, Springer, 1994, 242-411. Zbl0814.60093
  20. [20] M. F. Norman, Markov Processes and Learning Models, Academic Press, New York, 1972. Zbl0262.92003
  21. [21] Y. Peres, A combinatorial application of the maximal ergodic theorem, Bull. London Math. Soc. 20 (1988), 248-252. Zbl0642.10051
  22. [22] Ya. G. Sinai, Construction of Markov partitions, Funktsional. Anal. i Prilozhen. 2 (1968), no. 3, 70-80 (in Russian). 
  23. [23] Ya. G. Sinai, The limiting behaviour of a one-dimensional random walk in a random environment, Theory Probab. Appl. 27 (1982), 256-268. 
  24. [24] Ya. G. Sinai, Simple random walks on tori, preprint. 
  25. [25] R. Sine, On invariant probabilities for random rotations, Israel J. Math. 33 (1979), 384-388. Zbl0435.60075
  26. [26] F. Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), 1-31. Zbl0305.60029

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.