Invariance principle for the random conductance model in a degenerate ergodic environment

Abstract

We study a continuous time random walk, X, on Z d in an environment of random conductances taking values in (0, ∞). We assume that the law of the conductances is ergodic with respect to space shifts. We prove a quenched invariance principle for X under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme.

Gaussian bounds and parabolic Harnack inequality on locally irregular graphs

“…A second difficulty is that our proof of the global 'long range' heat kernel lower bound in Lemma 6.2 does not work for the VSRW. We remark that in [1,9], where Gaussian bounds are proved for the transition density of both the CSRW and the VSRW, stronger hypotheses are needed for the VSRW.…”

“…The ideas in [13], and more generally the methods of Moser and Nash on which [13] is based, have proved very fruitful in the study of random walks in symmetric random environments. For example [4] used Nash's ideas to obtain Gaussian bounds on the heat kernel on supercritical percolation clusters in Z d , while [1,2] use Moser's iteration argument to obtain a quenched invariance principle and Harnack inequalities for the random conductance model under quite general conditions.…”

“…Proof of Theorem 5 1. We can assume that h > λ 2 R 0 , otherwise, the left side of the weighted PI is empty.Let f : V → R and write f i = f B * i .…”

“…For the proof see [6,Proposition 3.3], and particularly equation (3.9). Fix t 1 ∈ [ 1 4 T, 1 2 T ], t 2 ∈ [ 3 4 T, T ] and y 1 , y 2 ∈ B 0 . Using (1.12), we get for z ∈ B 1 ,…”

Quenched invariance principle for random walks with time-dependent ergodic degenerate weights

“…During the last decade, considerable effort has been invested in the derivation of a quenched functional central limit theorem (QFCLT) or quenched invariance principle, see the surveys [13,27] (and references therein), and [4,11,22] for more recent results on the static RCM. For the time-dynamic RCM with ergodic degenerate conductances the following QFCLT has been shown in [3].…”