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Journal of Theoretical Probability

, Volume 24, Issue 3, pp 634–656 | Cite as

Limit Law of the Local Time for Brox’s Diffusion

  • Pierre Andreoletti
  • Roland DielEmail author
Article

Abstract

We consider Brox’s model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t,m log t +x)/t, xR), where m log t is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case for which the same questions have been answered recently by Gantert, Peres, and Shi (Ann. Inst. Henri Poincaré, Probab. Stat. 46(2):525–536, 2010).

Keywords

Diffusion process in Brownian potential Local time 

Mathematics Subject Classification (2000)

60J25 60J55 

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References

  1. 1.
    Bertoin, J.: Spliting at the infimum and excursions in half-lines for random walks and Levy processes. Stoch. Process. Appl. 47, 17–35 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Brox, T.: A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14(4), 1206–1218 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Carmona, P.: The mean velocity of a Brownian motion in a random Lévy potential. Ann. Probab. 25(4), 1774–1788 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cheliotis, D.: Localization of favorite points for diffusion in random environment. Stoch. Process. Appl. 118(7), 1159–1189 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Devulder, A.: Almost sure asymptotics for a diffusion process in a drifted Brownian potential. Preprint Google Scholar
  6. 6.
    Devulder, A.: The maximum of the local time of a diffusion in a drifted Brownian potential. Preprint Google Scholar
  7. 7.
    Gantert, N., Peres, Y., Shi, Z.: The infinite valley for a recurrent random walk in random environment. Ann. Inst. Henri Poincaré, Probab. Stat. 46(2), 525–536 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Golosov, A.O.: Localization of random walks in one-dimensional random environments. Commun. Math. Phys. 92, 491–506 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Golosov, A.O.: On limiting distribution for a random walk in a critical one-dimensional random environment. Commun. Mosc. Math. Soc. 41, 189–190 (1986) MathSciNetGoogle Scholar
  10. 10.
    Hu, Y., Shi, Z.: The limits of Sinai’s simple random walk in random environment. Ann. Probab. 26(4), 1477–1521 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hu, Y., Shi, Z.: The local time of simple random walk in random environment. J. Theor. Probab. 11(3), 765–793 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hu, Y., Shi, Z., Yor, M.: Rates of convergence of diffusions with drifted Brownian potentials. Trans. Am. Math. Soc. 351(10), 3915–3934 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kawazu, K., Tanaka, H.: A diffusion process in a Brownian environment with drift. J. Math. Soc. Jpn. 49, 189–211 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kesten, H.: The limit distribution of Sinai’s random walk in random environment. Physica A 138, 299–309 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Le Gall, J.F.: Sur la mesure de Hausdorff de la courbe Brownienne. In: Sém. Probability XIX. Lecture Notes in Math., vol. 1123, pp. 297–313. Springer, Berlin (1985) CrossRefGoogle Scholar
  16. 16.
    Neveu, J., Pitman, J.: Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. In: Séminaire de Probabilitées xxiii. Lecture Notes Math., vol. 1372, pp. 239–247. Springer, Berlin (1989) CrossRefGoogle Scholar
  17. 17.
    Révész, P.: Random Walk in Random and Non-random Environments. World Scientific, Singapore (1989) Google Scholar
  18. 18.
    Schumacher, S.: Diffusions with random coefficients. Contemp. Math. 41, 351–356 (1985) MathSciNetGoogle Scholar
  19. 19.
    Shi, Z.: A local time curiosity in random environment. Stoch. Process. Appl. 76(2), 231–250 (1998) zbMATHCrossRefGoogle Scholar
  20. 20.
    Shi, Z.: Sinai’s walk via stochastic calculus. Panor. Synth. 12, 53–74 (2001) Google Scholar
  21. 21.
    Sinai, Y.G.: The limit behaviour of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27(2), 256–268 (1982) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Singh, A.: Rates of convergence of a transient diffusion in a spectrally negative Lévy potential. Ann. Probab. 36(3), 279–318 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Solomon, F.: Random walks in random environment. Ann. Probab. 3(1), 1–31 (1975) zbMATHCrossRefGoogle Scholar
  24. 24.
    Taleb, M.: Large deviations for a Brownian motion in a drifted Brownian potential. Ann. Probab. 29(3), 1173–1204 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Tanaka, H.: Limit theorem for one-dimensional diffusion process in Brownian environment. In: Lecture Notes in Mathematics, vol. 1322. Springer, Berlin (1988) Google Scholar
  26. 26.
    Tanaka, H.: Localization of a diffusion process in a one-dimensional Brownian environment. Commun. Pure Appl. Math. 17, 755–766 (1994) CrossRefGoogle Scholar
  27. 27.
    Tanaka, H.: Limit theorem for a Brownian motion with drift in a white noise environment. Chaos Solitons Fractals 11, 1807–1816 (1997) CrossRefGoogle Scholar
  28. 28.
    Williams, D.: Path decomposition and continuity of local time for one dimensional diffusions. Proc. Lond. Math. Soc. 28, 738–768 (1974) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Laboratoire MAPMO–C.N.R.S. UMR 6628–Fédération Denis-PoissonUniversité d’OrléansOrléansFrance

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