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Variance decay for functionals of the environment viewed by the particle

Jean-Christophe Mourrat — 2011

Annales de l'I.H.P. Probabilités et statistiques

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,...

Scaling limit of the random walk among random traps on ℤd

Jean-Christophe Mourrat — 2011

Annales de l'I.H.P. Probabilités et statistiques

Attributing a positive value to each ∈ℤ, we investigate a nearest-neighbour random walk which is reversible for the measure with weights ( ), often known as “Bouchaud’s trap model.” We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that ≥5. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing...

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