Advertisement

On Random Walks in Random Environment on Trees and Their Relationship with Multiplicative Chaos

  • Mikhail Menshikov
  • Dimitri Petritis
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

The purpose of this paper is to report on recent results concerning random walks in a random environment on monochromatic and coloured trees and their relationship with multiplicative chaos. The proofs are omitted since they are extensively given elsewhere [12]. It is worth noticing that for the random walk on monochromatic tree the results we give were previously known [11]; we provide however a totally new proof, based solely on multiplicative chaos results, that allows to relax some stringent conditions on independence properties of the random transition probabilities. For the random walk on a coloured tree the results are new; the classification of the asymptotic behaviour of the random walk allows to obtain some hints for the classification of the yet unsolved corresponding multiplicative chaos problem.

Keywords

Random Walk Random Environment Random String Coloured Tree Reversible Markov Chain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. Ben Nasr. Mesures aléatoires de Mandelbrot associées à des substitutions. C. Rendus Acad. Sciences (Paris) 304:253–258, 1987.Google Scholar
  2. [2]
    Brigitte Chauvin. Martingales produits et lignes d’arrêt pour un processus de branchement brownien. C. R. Acad. Sci. Paris Sér. I Math. 306(12):499–502, 1988.MathSciNetzbMATHGoogle Scholar
  3. [3]
    P. Collet and F. Koukiou. Large deviations for multiplicative chaos. Commun. Math. Phys. 147:329–342, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    F. Comets, M. Menshikov, and S. Popov. Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26:1433–1445, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    R. Durrett and Th. Liggett. Fixed points of smoothing transformation. Z. Wahrscheinlichkeitstheorie verw. Gebiete 64:275–301, 1983.Google Scholar
  6. [6]
    Y. Guivarc’h. Sur une extension de la notion semi-stable. Ann. Inst. H. Poicaré 26:261–285, 1990.MathSciNetzbMATHGoogle Scholar
  7. [7]
    J.-P. Kahane and J. Peyrière. Sur certaines martingales de Benoît Mandelbrot. Adv. Math. 22:131–145, 1976.zbMATHGoogle Scholar
  8. [8]
    Q. Liu. Sur une équation fonctionnelle et ses applications: une extension du théorème de Kesten-Stigun concernant les processus de branchement. Ann. Appl. Probab. 29:353–373, 1997.zbMATHCrossRefGoogle Scholar
  9. [9]
    Q. Liu. Fixed points of a generalised smoothing transformation and applications to the branching random walk. Ann. Appl. Probab. 30:85–112, 1998.zbMATHCrossRefGoogle Scholar
  10. [10]
    Quansheng Liu and Alain Rouault. Limit theorems for Mandelbrot’s multiplicative cascades. Ann. Appl. Probab. 10(1):218–239, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    R. Lyons and R. Pemantle. Random walk in a random environment and first passage percolation on trees. Ann. Probab. 20:125–136, 1991.MathSciNetCrossRefGoogle Scholar
  12. [12]
    M. Menshikov and D. Petritis. Random walks in random environment on trees and multiplicative chaos, preprint 2001 submitted for publication, eprint arXiv:math.PR/0112103.Google Scholar
  13. [13]
    M. Menshikov, D. Petritis, and S. Popov. Matrix multiplicative chaos and bindeweeds, in preparation 2002.Google Scholar
  14. [14]
    E. Waymire and S. Williams. A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 348:585–632, 1996.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Mikhail Menshikov
    • 1
  • Dimitri Petritis
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DurhamUK
  2. 2.IRMARUniversitéde Rennes IRennes CedexFrance

Personalised recommendations