Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
An almost sure invariance principle for random walks in a space-time random environment
Download PDF
Download PDF
  • Published: 10 February 2005

An almost sure invariance principle for random walks in a space-time random environment

  • Firas Rassoul-Agha1 &
  • Timo Seppäläinen2 

Probability Theory and Related Fields volume 133, pages 299–314 (2005)Cite this article

  • 262 Accesses

  • 50 Citations

  • Metrics details

Abstract.

We consider a discrete time random walk in a space-time i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an L2 averaged drift. We also state an a.s. invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Basu, A.K.: A functional central limit theorem for Banach space valued dependent variables and a log log law. Ser. A 49, 275–287 (1987)

    Google Scholar 

  2. Bérard, J.: The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment. J. Appl. Probab. 41, 83–92 (2004)

    Article  Google Scholar 

  3. Bernabei, M.S., Boldrighini, C., Minlos, R.A., Pellegrinotti, A.: Almost-sure central limit theorem for a model of random walk in fluctuating random environment. Markov Process. Related Fields. 4, 381–393 (1998)

    Google Scholar 

  4. Boldrighini, C., Minlos, R.A., Pellegrinotti, A.: Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive. Probab. Theory Relat. Fields. 129, 133–156 (2004)

    Google Scholar 

  5. Boldrighini, C., Minlos, R.A., Pellegrinotti, A.: Random walk in a fluctuating random environment with Markov evolution. In On Dobrushin's way. From probability theory to statistical Physics. AMS Transl. Ser. 198, Am. Math. Soc. Providence, RI, 2000, pp. 13–35

  6. Boldrighini, C., Minlos, R.A., Pellegrinotti, A.: Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment. Probab. Theory Relat. Fields. 109, 245–273 (1997)

    Article  Google Scholar 

  7. Bolthausen, E., Sznitman, A-S.: On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9, 345–375 (2002)

    Google Scholar 

  8. Bolthausen, E., Sznitman, A-S., Zeitouni, O.: Cut points and diffusive random walks in random environment. Ann. Inst. H. Poincaré Probab. Statist. 39, 527–555 (2003)

    Article  Google Scholar 

  9. Bricmont, J., Kupiainen, A.: Random walks in asymmetric random environments. Commun. Math. Phys. 142, 345–420 (1991)

    Article  Google Scholar 

  10. Comets, F., Zeitouni, O.: Gaussian fluctuations for random walks in random mixing environments. Preprint, 2004

  11. Derriennic, Y., Lin, M.: Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123, 93–130 (2001)

    Google Scholar 

  12. Derriennic, Y., Lin, M.: The central limit theorem for Markov chains started at a point. Probab. Theory Relat. Fields. 125, 73–76 (2003)

    Article  Google Scholar 

  13. Durrett, R.: Probability: Theory and examples. Duxbury Press, Belmont, CA, 1996

  14. Ethier, S.N., Kurtz, T.G.: Markov processes: Characterization and convergence. John Wiley & Sons Inc., New York, 1986

  15. Ferrari, P.A., Fontes, L.R.G.: Fluctuations of a surface submitted to a random average process. Electron. J. Probab. 3, 1–34 (1998)

    Google Scholar 

  16. Kipnis, C., Varadhan, S.R.S.: A central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Commun. Math. Phys. 104, 1–19 (1986)

    Article  Google Scholar 

  17. Komorowski, T., Olla, S.: A note on the central limit theorem for two-fold stochastic random walks in a random environment. Bull. Polish Acad. Sci. Math. 51, 217–232 (2003)

    Google Scholar 

  18. Lawler, G.F.: Weak convergence of a random walk in a random environment. Commun. Math. Phys. 87, 81–87 (1982)

    Article  Google Scholar 

  19. Maxwell, M., Woodroofe, M.: Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28, 713–724 (2000)

    Article  Google Scholar 

  20. Olla, S.: Notes on the central limit theorems for tagged particles and diffusions in random fields. Given at Etàts de la recherche: Milieux Alèatoires. Panorama et Synthèses. 12, 75–100 (2001)

    Google Scholar 

  21. Rassoul-Agha, F., Seppäläinen, T.: Ballistic random walk in random environment with a forbidden direction. Preprint, 2004

  22. Rosenblatt, M.: Markov processes. Structure and asymptotic behavior. Springer-Verlag, New York, 1971

  23. Sidoravicius, V., Sznitman, A-S.: Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Relat. Fields. 129, 219–244 (2004)

    Article  Google Scholar 

  24. Spitzer, F.: Principles of random walks. Springer-Verlag, Berlin-Heidelberg-New York, 1976

  25. Stannat, W.: A remark on the CLT for a random walk in a random environment. Probab. Theory Relat. Fields. Published Online, 2004

  26. Sznitman, A-S.: An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Relat. Fields. 122, 509–544 (2002)

    Article  Google Scholar 

  27. Sznitman, A-S., Zeitouni, O.: An invariance principle for isotropic diffusions in random environments. Preprint, 2004

  28. Tóth, B.: Persistent random walks in random environment. Probab. Theory Relat. Fields. 71, 615–625 (1986)

    Article  Google Scholar 

  29. Zeitouni, O.: Random walks in random environments. Lecture Notes in Mathematics 1837, Springer-Verlag, Berlin, pp. 189–312, 2004

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Ohio State University, 231 West 18th Avenue, Columbus, OH, 43210, USA

    Firas Rassoul-Agha

  2. Mathematics Department, University of Wisconsin-Madison, Van Vleck Hall, Madison, WI, 53706, USA

    Timo Seppäläinen

Authors
  1. Firas Rassoul-Agha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Timo Seppäläinen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Firas Rassoul-Agha.

Additional information

T. Seppäläinen was partially supported by National Science Foundation grant DMS-0402231.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rassoul-Agha, F., Seppäläinen, T. An almost sure invariance principle for random walks in a space-time random environment. Probab. Theory Relat. Fields 133, 299–314 (2005). https://doi.org/10.1007/s00440-004-0424-1

Download citation

  • Received: 30 June 2004

  • Revised: 20 November 2004

  • Published: 10 February 2005

  • Issue Date: November 2005

  • DOI: https://doi.org/10.1007/s00440-004-0424-1

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Discrete Time
  • Invariant Measure
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature