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# Variance limite d'une marche aléatoire réversible en milieu aléatoire sur ZZ

(Impact Factor: 0.47). 04/2009; 347(s 7–8):401–406. DOI: 10.1016/j.crma.2009.01.030
Source: arXiv

ABSTRACT

The Central Limit Theorem for the random walk on a stationary random network of conductances has been studied by several authors. In one dimension, when conductances and resistances are integrable, and following a method of martingale introduced by S. Kozlov (1985), we can prove the Quenched Central Limit Theorem. In that case the variance of the limit law is not null. When resistances are not integrable, the Annealed Central Limit Theorem with null variance was established by Y. Derriennic and M. Lin (personal communication). The quenched version of this last theorem is proved here, by using a very simple method. The similar problem for the continuous diffusion is then considered. Finally our method allows us to prove an inequality for the quadratic mean of a diffusion. To cite this article: J. Depauw, J.-M. Derrien, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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ABSTRACT: The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple andwas introduced by Depauw and Derrien [3]. More precisely, for a given realizationω of the environment, we consider the Poisson equation (Pω-I)g = f , and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.
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