Quenched invariance principle for random walks among random degenerate conductances

Abstract

We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik [Ann. Probab.] and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.

Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations

“…In this section, we show that the results of Section 3 can be used to weaken the assumption of [12,18] in order to establish L I -sublinearity of the corrector. The application to the quenched invariance principle for the random walk in a random degenerate environment can be found in [7]. Let us now be more precise and phrase the assumptions on the coefficient fields by appealing to the language of ergodic, measure-preserving dynamical systems (which is a standard in the theory of stochastic homogenization; see, e.g., the seminal paper [30] We omit the proof of Lemma 5.2 since it is by now standard.…”

“…In this section, we show that the results of Section 3 can be used to weaken the assumption of [12,18] in order to establish L I -sublinearity of the corrector. The application to the quenched invariance principle for the random walk in a random degenerate environment can be found in [7].…”

Quenched tail estimate for the random walk in random scenery and in random layered conductance

“…The second part of Theorem 1.4 (and Remark 1.5) shows that (1.23) can be relaxed to E[z(0)] < ∞ in our special model. More recently in [3], the invariance principle and the elliptic Harnack inequality are proved under a slightly weaker condition where Finally, let us recall the representation of the process ((X t ) t≥0 , (P ω x ) x∈Z 1+d ) in terms of the random walk in random scenery, which is a key ingredient in the proof. Let ((S 1 t , S 2 t ) t≥0 , (P x ) x∈Z 1+d ) be the continuous time simple random walk on Z 1+d which jumps to each of the neighboring sites at rate one, where S 1 is the first onedimensional component and S 2 is the remaining d-dimensional component.…”

Higher integrability for variational integrals with non-standard growth

“…3. This idea has its origin in joint works with Bella [4,5] on linear non-uniformly elliptic equations.…”