The Annals of Probability
- Ann. Probab.
- Volume 33, Number 5 (2005), 1760-1781.
Diffusion in random environment and the renewal theorem
According to a theorem of Schumacher and Brox, for a diffusion X in a Brownian environment, it holds that (Xt−blogt)/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.
Ann. Probab., Volume 33, Number 5 (2005), 1760-1781.
First available in Project Euclid: 22 September 2005
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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