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We study continuous time random walks on $\mathbb{Z}^d$ (with $d \geq 2$) among random conductances $\{ \omega(\{x,y\}) : x,y \in \mathbb{Z}^d\}$ that permit jumps of arbitrary length. The law of the random variables $\omega(\{x,y\})$, taking values in $[0, \infty)$, is assumed to be stationary and ergodic with respect to space shifts. Assuming that the first moment of $\sum_{x \in \mathbb{Z}^d} \omega(\{0,x\}) |x|^2$ and the $q$-th moment of $1/\omega(0,x)$ for $x$ neighbouring the origin are finite for some $ q >d/2$, we show a quenched invariance principle and a quenched local limit theorem, where the moment condition is optimal for the latter. We also obtain H\"older regularity estimates for solutions of the heat equation for the associated non-local discrete operator, and deduce that the pointwise spectral dimension equals $d$ almost surely. Our results apply to random walks on long-range percolation graphs with connectivity exponents larger than $d+2$ when all nearest-neighbour edges are present., Comment: There is a gap in the proof of Proposition 4.2, step 3

We present on-diagonal heat kernel estimates and quantitative homogenization statements for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using standard techniques, with key inputs coming from a careful analysis of the volume growth of the invariant measure of the process under study. As for the quantitative homogenization results, these include both quenched and annealed Berry-Esseen-type theorems, as well as a quantitative quenched local limit theorem. Whilst the model we study here is a particularly simple example of a random walk in a random environment, we believe the roadmap we provide for establishing the latter result in particular will be useful for deriving quantitative local limit theorems in other, more challenging, settings., Comment: 33 pages, accepted version, to appear in Stochastic Process. Appl

In this paper we consider a time-continuous random walk in $\mathbb{Z}^d$ in a dynamical random environment with symmetric jump rates to nearest neighbours. We assume that these random conductances are stationary and ergodic and, moreover, that they are bounded from below but unbounded from above with finite first moment. We derive sharp on-diagonal estimates for the annealed first and second discrete space derivative of the heat kernel which then yield local limit theorems for the corresponding kernels. Assuming weak algebraic off-diagonal estimates, we then extend these results to the annealed Green function and its first and second derivative. Our proof which extends the result of Delmotte and Deuschel (2005) to unbounded conductances with first moment only, is an adaptation of the recent entropy method of Benjamini et. al. (2015)., Comment: 37 pages

We establish a quenched local central limit theorem for the dynamic random conductance model on $\mathbb{Z}^d$ only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show H\"older 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., Comment: 33 pages, accepted version, to appear in Probab. Theory Relat. Fields

We consider a stationary and ergodic random field $\{\omega(e) : e \in E_d\}$ that is parameterized by the edge set of the Euclidean lattice $\mathbb{Z}^d$, $d \geq 2$. The random variable $\omega(e)$, taking values in $[0, \infty)$ and satisfying certain moment bounds, is thought of as the conductance of the edge $e$. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster $\mathcal{C}_{\infty}(\omega)$, we prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infinite cluster. An essential ingredient of our proof is a new anchored relative isoperimetric inequality., Comment: 22 pages