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.
We study a continuous time random walk X in an environment of dynamic random conductances in Z d . We assume that the conductances are stationary ergodic, uniformly bounded and bounded away from zero and polynomially mixing in space and time. We prove a quenched invariance principle for X, and obtain Green's functions bounds and a local limit theorem. We also discuss a connection to stochastic interface models.
ABSTRACT. We establish elliptic and parabolic Harnack inequalities on graphs with unbounded weights. As an application we prove a local limit theorem for a continuous time random walk X in an environment of ergodic random conductances taking values in (0, ∞) satisfying some moment conditions.
Let (X , d, m) be a metric measure space with a local regular Dirichlet form. We establish necessary and sufficient conditions for upper heat kernel bounds with subdiffusive space-time exponent to hold. This characterization is stable under rough isometries, that is it is preserved under bounded perturbations of the Dirichlet form. Further, we give a criterion for stochastic completeness in terms of a Sobolev inequality for cutoff functions. As an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space, where the well-established criterion in terms of volume growth fails.
We establish a quenched local central limit theorem for the dynamic random conductance model on Z d only assuming ergodicity with respect to spacetime shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi's iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate ergodic weights.
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