Variance decay for functionals of the environment viewed by the particle

By Jean-Christophe Mourrat. 2009

72 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1ee8243a165ef5a26d974da79af0a8a9c7a1609a

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Citations in the corpus

Quantitative estimates in stochastic homogenization for correlated coefficient fields (2019, background): “ptimal estimates on the corrector, the ﬂuctuations of the energy density of the corrector, and the approximation of homogenized coeﬃcients in [22, 23, 25]. Using more probabilistic arguments, Mourrat [31] obtained suboptimal estimates on the decay of the associated semi-group (the environment as see”
Lipschitz Regularity for Elliptic Equations with Random Coefficients (2014, unknown): “Expounding on this work and some ideas developed independently in [32], these authors and Neukamm [21, 22] then proved optimal estimates and a quantitative two-scale expansion for uniformly elliptic equations under quantitative assumptions built on spectral gap-type concentration inequalities.”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “Previously, the first quantitative (but suboptimal) estimates on the semigroup were obtained in [93].”
Quantitative stochastic homogenization of convex integral functionals (2014, background): “[22] J.-C. Mourrat and J. Nolen.”
Quantitative results on the corrector equation in stochastic homogenization (2014, methodology): “A first possibility to avoid the use of a covariance estimate is to appeal to spectral theory (in the case of symmetric coefficients) to bound this covariance using the variance of ψT , in the spirit of [23, 11] in the discrete setting (see also Corollary 2).”
Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics (2013, methodology): “As already used by Mourrat in [32], estimates on the semigroup allow one to quantify the bottom of the spectrum of D∗a(0)D.”
An optimal variance estimate in stochastic homogenization of discrete elliptic equations (2011, methodology): “ls, there exists some nonnegative measure ed associated with the elliptic operator and direction ξ∈Rd, such that the homogenized coeﬃcient is given by ξ·A homξ=hξ·Aξi− Z R+ 1 λ ded(λ). As recalled in [11], we also have ξ·ATξ=hξ·Aξi− Z R+ λ−2/T (1/T+λ)2 (1.11) ded(λ). In particular, the systematic er”
A Regularity Theory for Random Elliptic Operators (2014, background): “timal estimates on the corrector, the uctuations of the energy density of the corrector, and the approximation of homogenized coecients in [21, 22, 24]. Using more probabilistic arguments, 2 Mourrat [29] independently obtained suboptimal estimates on the decay of the associated semi-group (the envir”
A higher-order large-scale regularity theory for random elliptic operators (2015, background): “For a more probabilistic viewpoint of stochastic homogenization of linear elliptic equations, see [15].”