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September 2005 Diffusion in random environment and the renewal theorem
Dimitrios Cheliotis
Ann. Probab. 33(5): 1760-1781 (September 2005). DOI: 10.1214/009117905000000279


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.


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Dimitrios Cheliotis. "Diffusion in random environment and the renewal theorem." Ann. Probab. 33 (5) 1760 - 1781, September 2005.


Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1083.60081
MathSciNet: MR2165578
Digital Object Identifier: 10.1214/009117905000000279

Primary: 60K37
Secondary: 60G17 , 60J65

Keywords: Brownian motion , diffusion , favorite point , random environment , Renewal theorem , Sinai’s walk

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 5 • September 2005
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