The Annals of Probability

Diffusion in random environment and the renewal theorem

Dimitrios Cheliotis

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According to a theorem of Schumacher and Brox, for a diffusion X in a Brownian environment, it holds that (Xtblogt)/log2t→0 in probability, as t→∞, where b is a stochastic process having an explicit description and depending only on the environment. We compute the distribution of the number of sign changes for b on an interval [1,x] and study some of the consequences of the computation; in particular, we get the probability of b keeping the same sign on that interval. These results have been announced in 1999 in a nonrigorous paper by Le Doussal, Monthus and Fisher [Phys. Rev. E 59 (1999) 4795–4840] and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. Finally, we comment on the information these results give about the behavior of the diffusion.

Article information

Ann. Probab., Volume 33, Number 5 (2005), 1760-1781.

First available in Project Euclid: 22 September 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 60G17: Sample path properties 60J65: Brownian motion [See also 58J65]

Diffusion random environment renewal theorem Brownian motion Sinai’s walk favorite point


Cheliotis, Dimitrios. Diffusion in random environment and the renewal theorem. Ann. Probab. 33 (2005), no. 5, 1760--1781. doi:10.1214/009117905000000279.

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  • Brox, Th. (1986). A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14 1206–1218.
  • Comets, F. and Popov, S. (2003). Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Related Fields 126 571–609.
  • Dembo, A., Guionnet, A. and Zeitouni, O. (2001). Aging properties of Sinai's model of random walk in random environment. In Saint Flour Summer School 2001 Lecture Notes (O. Zeitouni, ed.). Available at
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Wadsworth, Pacific Grove, CA.
  • Golosov, A. O. (1984). Localization of random walks in one-dimensional random environments. Comm. Math. Phys. 92 491–506.
  • Hu, Y. (2000). Tightness of localization and return time in random environment. Stochastic Process. Appl. 86 81–101.
  • Kesten, H. (1986). The limit distribution of Sinai's random walk in random environment. Phys. A 138 299–309.
  • Krantz, S. (1992). Function Theory of Several Complex Variables. Wadsworth, Pacific Grove, CA.
  • Le Doussal, P., Monthus, C. and Fisher, D. (1999). Random walkers in one-dimensional random environments: Exact renormalization group analysis. Phys. Rev. E 59 4795–4840.
  • Marsden, J. and Hoffman, M. (1987). Basic Complex Analysis, 2nd ed. W. H. Freeman and Company, New York.
  • Neveu, J. and Pitman, J. (1989). Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 239–247. Springer, Berlin.
  • Schumacher, S. (1984). Diffusions with random coefficients. Ph.D. dissertation, UCLA.
  • Schumacher, S. (1985). Diffusions with random coefficients. In Particle Systems, Random Media and Large Deviations (R. Durett, ed.) 351–356. Amer. Math. Soc., Providence, RI.
  • Seignourel, P. (2000). Discrete schemes for processes in random media. Probab. Theory Related Fields 118 293–322.
  • Shi, Z. (2001). Sinai's walk via stochastic calculus. In Milieux Aléatoires, Panoramas et Synthèses (F. Comets and E. Pardoux, eds.) 12 53–74. Société Mathématique de France.
  • Sinai, Y. G. (1982). The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 256–268.
  • Tanaka, H. (1987). Limit distribution for 1-dimensional diffusion in a reflected Brownian medium. Séminaire de Probabilités XXI. Lecture Notes in Math. 1247 246–261. Springer, New York.
  • Tanaka, H. (1987). Limit distributions for one-dimensional diffusion processes in self-similar random environments. In Hydrodynamic Behavior and Interacting Particle Systems (G. Papanicolaou, ed.) 189–210. Springer, New York.
  • Tanaka, H. (1988). Limit theorem for one-dimensional diffusion process in Brownian environment. Stochastic Analysis. Lecture Notes in Math. 1322 157–172. Springer, New York.
  • Tanaka, H. (1994). Localization of a diffusion process in a one-dimensional Brownian environment. Comm. Pure Appl. Math. 47 755–766.