Variance Decay for Functionals of the Environment Viewed by the Particle

For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. As an example of application, we show that under certain conditions, our results imply an estimate of the speed of convergence of the mean square displacement of the walk towards its limit.