Skip to main content
SpringerLink
Log in

Cut Points and Diffusions in Random Environment

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

In this article we investigate the asymptotic behavior of a new class of multidimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen et al. (Ann. Inst. Henri Poincaré 39(5):527–555, 2003) in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berger, N., Zeitouni, O.: A quenched invariance principle for certain ballistic random walks in i.i.d. environments (2008). Available at: arXiv:math/0702306v3

  2. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  3. Bolthausen, E., Sznitman, A.-S.: On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9(3), 345–376 (2002)

    MATH  MathSciNet  Google Scholar 

  4. Bolthausen, E., Sznitman, A.-S., Zeitouni, O.: Cut points and diffusive random walks in random environment. Ann. Inst. Henri Poincaré 39(5), 527–555 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Comets, F., Zeitouni, O.: A law of large numbers for random walks in random mixing environments. Ann. Probab. 32(1b), 880–914 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Deuschel, J.-D., Stroock, D.W.: Large Deviations. Am. Math. Soc., Providence (2001)

    Google Scholar 

  8. Durrett, R.: Probability: Theory and Examples, 3rd edn. Duxbury, Belmont (2005)

    Google Scholar 

  9. Ethier, S.N., Kurtz, T.G.: Markov Processes. Wiley, New York (1986)

    Book  MATH  Google Scholar 

  10. Goergen, L.: An effective criterion and a new example for ballistic diffusions in random environment. Ann. Probab. (2008, to appear). Also available at: arXiv:0706.4069v1

  11. Gut, A.: Probability: A Graduate Course. Springer, New York (2005)

    MATH  Google Scholar 

  12. Ilin, A.M., Kalashnikov, A.S., Oleinik, O.A.: Linear equations of the second order of parabolic type. Russ. Math. Surv. 17(3), 1–143 (1962)

    Article  Google Scholar 

  13. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)

    MATH  Google Scholar 

  14. Komorowski, T., Krupa, G.: On the existence of invariant measure for Lagrangian velocity in compressible environments. J. Stat. Phys. 106(3–4), 635–651 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Komorowski, T., Krupa, G.: On stationarity of Lagrangian observations of passive tracer velocity in a compressible environment. Ann. Appl. Probab. 14(4), 1666–1697 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Komorowski, T., Olla, S.: On homogenization of time-dependent random flows. Probab. Theory Relat. Fields 121(1), 98–116 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lawler, G.F.: Intersection of Random Walks. Birkhäuser, Boston (1991)

    Google Scholar 

  18. Lawler, G.F.: Cut times for Brownian motion and random walk. Paul Erdös and his mathematics I. Bolyai Soc. Math. Stud. 11, 411–421 (2002)

    MathSciNet  Google Scholar 

  19. Lyons, T.J., Zheng, W.A.: On conditional diffusion processes. Proc. R. Soc. Edinb. 115A, 243–255 (1990)

    MathSciNet  Google Scholar 

  20. Neveu, J.: Processus ponctuel. In: Dold, A., Eckmann, B. (eds.) École d’Été de Probabilités de Saint-Flour. Lecture Notes in Mathematics, vol. 598, pp. 250–445. Springer, Berlin (1977)

    Google Scholar 

  21. Osada, H.: Homogenization of diffusion processes with random stationary coefficients. In: Dold, A., Eckmann, B. (eds.) Probability Theory and Mathematical Statistics. Lecture Notes in Mathematics, vol. 1021, pp. 507–517. Springer, Berlin (1983)

    Chapter  Google Scholar 

  22. Protter, P.: Stochastic Integration and Differential Equations, 3rd edn. Springer, Berlin (1990)

    MATH  Google Scholar 

  23. 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)

    Article  MATH  Google Scholar 

  24. Rassoul-Agha, F., Seppäläinen, T.: Almost sure functional central limit theorem for ballistic random walk in random environment. Ann. Inst. Henri Poincaré (2008, to appear). Also available at: arXiv:0705.4116v3

  25. Rassoul-Agha, F., Seppäläinen, T.: Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction. Ann. Probab. 35(1), 1–31 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schmitz, T.: Diffusions in random environment with ballistic behavior. Ann. Inst. Henri Poincaré Probab. Stat. 42(6), 683–714 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Schmitz, T.: Examples of condition (T) for diffusions in random environment. Electron. J. Probab. 11, 540–562 (2006)

    MathSciNet  Google Scholar 

  28. Shen, L.: On ballistic diffusions in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 39(5), 839–876 (2004). Addendum in Ann. Inst. Henri Poincaré. Probab. Stat. 40(3), 385–386

    Article  Google Scholar 

  29. Stroock, D.W.: Probability Theory, An Analytic View. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  30. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)

    MATH  Google Scholar 

  31. Sznitman, A.-S.: On a class of transient random walks in random environment. Ann. Probab. 29(2), 723–765 (2001)

    Article  MathSciNet  Google Scholar 

  32. Sznitman, A.-S.: An effective criterion for ballistic random walks in random environment. Probab. Theory Relat. Fields 122(4), 509–544 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sznitman, A.-S.: Topics in random walk in random environment. In: Lawler, G.F. (ed.) School and Conference on Probability Theory, May 2002. ICTP Lecture Series, vol. 17, pp. 203–266. Int. Cent. Theor. Phys., Trieste (2004)

    Google Scholar 

  34. Sznitman, A.-S.: On new examples of ballistic random walks in random environment. Ann. Probab. 31(1), 285–322 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Sznitman, A.-S.: Random motions in random media. In: Bovier, A., Dalibard, J., den Hollander, F., Dunlop, F., van Enter, A. (eds.) Mathematical Statistical Physics. Les Houches Session, vol. LXXXIII, pp. 219–242. Elsevier, Amsterdam (2005)

    Google Scholar 

  36. Sznitman, A.-S., Zeitouni, O.: An invariance principle for isotropic diffusions in random environment. Invent. Math. 164(3), 455–567 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Sznitman, A.-S., Zerner, M.P.W.: A law of large numbers for random walks in random environment. Ann. Probab. 27(4), 1851–1869 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  38. Whitt, W.: Convergence of probability measures on the function space C[0,∞). Ann. Math. Stat. 41(3), 939–944 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zeitouni, O.: Random walks in random environment. In: Picard, J. (ed.) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol. 1837, pp. 190–312. Springer, Berlin (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, ETH Zurich, 8092, Zurich, Switzerland

    Ivan del Tenno

Authors
  1. Ivan del Tenno
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan del Tenno.

Rights and permissions

Reprints and permissions

About this article

Cite this article

del Tenno, I. Cut Points and Diffusions in Random Environment. J Theor Probab 22, 891–933 (2009). https://doi.org/10.1007/s10959-008-0169-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-008-0169-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation