Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances

Andrés, Sebastián; Deuschel, Jean–Dominique; Slowik, Martin

doi:10.48550/arxiv.1711.11119

Cited by 2 publications

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“…The proofs of both Theorem 1.7 and the results in [24] follow the strategy in [28]. However, in [24] the required on-diagonal estimate on the heat kernel is derived from the anchored Nash inequality established in [45], which makes a uniform upper ellipticity necessary, while in our setting the analogue heat kernel bound in Lemma 2.5 can be deduced from the upper off-diagonal heat kernel estimates in [4,5].…”

Section: Resultsmentioning

confidence: 99%

“…On-diagonal heat kernel estimate: Proof of Lemma 2.5. The statement is a rather direct consequence of an on-diagonal estimate (see Lemma 2.9 below), which can be obtained from [5], and an application of the spectral gap estimate of Assumption 1.3 used to control moments of the estimate's random constant, see Lemma 2.10. Assuming M (p, q) < ∞ for any p, q ∈ (1, ∞), we denote by R = R(ω, p, q) ≥ 1 the smallest integer such that for all R ≥ R,…”

Section: Gradient Heat Kernel and Annealed Green's Function Estimatesmentioning

confidence: 99%

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Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

Andrés

Neukamm

2018

Stoch PDE: Anal Comp

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We study the random conductance model on the lattice Z d , i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension d ≥ 3 quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed t − 1 5 +ε for d ≥ 4 and t − 1 10 +ε for d = 3. Additionally, in the uniformly elliptic case in low dimensions d = 2, 3 we improve the rate in a quantitative Berry-Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for d ≥ 3 we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.1.1. The model. Let d ≥ 2. We study the nearest-neighbour random conductance model on the d-dimensional Euclidean lattice (Z d , E d ), where E d := {e = {x, x ± e i } : x ∈ Z d , i = 1, . . . , d } denotes the set of non-oriented nearest neighbour edges and {e 1 , . . . , e d } the canonical basis in R d . We endow the graph with positive random weights, which we describe by a family ω = {ω(e), e ∈ E d } ∈ Ω := (0, ∞) E d . We refer to ω(e) as the conductance of an edge e ∈ E d . To simplify notation, for any x, y ∈ Z d , we set ω(x, y) = ω(y, x) := ω({x, y}), ∀ {x, y} ∈ E d , ω(x, y) := 0, ∀ {x, y} ∈ E d , and define the matrix field ω : Z d → R d×d by ω(x) = diag(ω(x, x + e 1 ), . . . , ω(x, x + e d )). 2 1 5 if d = 2, c (t + 1) − 1 5 if d ≥ 3.5

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Section: Resultsmentioning

confidence: 99%

Section: Gradient Heat Kernel and Annealed Green's Function Estimatesmentioning

confidence: 99%

Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

Andrés

Neukamm

2018

Stoch PDE: Anal Comp

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“…The heat kernel has been object of very active research in recent years, see [14,7,10,9,8,16,11,5,6] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Green kernel asymptotics for two-dimensional random walks under random conductances

Andres,

Deuschel,

Slowik

2018

Preprint

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We consider random walks among random conductances on Z 2 and establish precise asymptotics for the associated potential kernel and the Green function of the walk killed upon exiting balls. The result is proven under a general set of assumptions, where examples include uniformly elliptic conductances, random walks on supercritical percolation clusters and ergodic degenerate conductances satisfying a moment condition. We also provide a similar result for the time-dynamic random conductance model. As an application we present a scaling limit for the variances in the Ginzburg-Landau ∇φ-interface model. CONTENTS1. Introduction 1 2. Green kernel asymptotics in the two-dimensional static RCM 6 2.1. Assumptions 6 2.2. Examples 8 2.3. Proof of Theorem 1.4 9 2.4. Proof of Theorem 1.5 10 3. Green kernel asymptotics in the two-dimensional dynamic RCM 12 3.1. Setting and results 12 3.2. Application to stochastic interface models 14 References 15

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Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances

Cited by 2 publications

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Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

Green kernel asymptotics for two-dimensional random walks under random conductances

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