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)

TL;DR

This paper investigates the variance behavior of reversible random walks in random environments on Z, establishing conditions for non-zero variance in the quenched CLT and providing new proofs and inequalities.

Contribution

It proves the quenched CLT with null variance when resistances are not integrable and introduces a simple method applicable to continuous diffusions.

Findings

Quenched CLT with non-zero variance for integrable conductances.

Quenched CLT with null variance for non-integrable resistances.

New inequality for quadratic mean of diffusions.

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 (X_t)_t at all time t.