The Annals of Probability

Diffusion in random environment and the renewal theorem

Dimitrios Cheliotis

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Abstract

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

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

Dates
First available in Project Euclid: 22 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1127395872

Digital Object Identifier
doi:10.1214/009117905000000279

Mathematical Reviews number (MathSciNet)
MR2165578

Zentralblatt MATH identifier
1083.60081

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

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

Citation

Cheliotis, Dimitrios. Diffusion in random environment and the renewal theorem. Ann. Probab. 33 (2005), no. 5, 1760--1781. doi:10.1214/009117905000000279. https://projecteuclid.org/euclid.aop/1127395872


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