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The Central Limit Theorem for the random walk on a stationary random networkof conductances has been studied by several authors. In one dimension, whenconductances and resistances are integrable, and following a method ofmartingale introduced by S. Kozlov (1985), we can prove the Quenched CentralLimit Theorem. In that case the variance of the limit law is not null. Whenresistances are not integrable, the Annealed Central Limit Theorem with nullvariance was established by Y. Derriennic and M. Lin (personal communication).The quenched version of this last theorem is proved here, by using a verysimple method. The similar problem for the continuous diffusion is thenconsidered. Finally our method allows us to prove an inequality for thequadratic mean of a diffusion (X_t)_t at all time t.

Variance limite d'une marche al\'eatoire r\'eversible en milieu  al\'eatoire sur Z (Limit of the Variance of a Reversible Random Walk in  Random Medium on Z)