Non-uniformly parabolic equations and applications to the random conductance model

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on $$\mathbb Z^d$$ Z d . In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.

Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

An improvement of convergence rate in the local limit theorem for integral-valued random variables

Let $X_{1}, X_{2}, \ldots , X_{n}$ X 1 , X 2 , … , X n be independent integral-valued random variables, and let $S_{n}=\sum_{j=1}^{n}X_{j}$ S n = ∑ j = 1 n X j . One of the interesting probabilities is the probability at a particular point, i.e., the density of $S_{n}$ S n . The theorem that gives the estimation of this probability is called the local limit theorem. This theorem can be useful in finance, biology, etc. Petrov (Sums of Independent Random Variables, 1975) gave the rate $O (\frac{1}{n} )$ O ( 1 n ) of the local limit theorem with finite third moment condition. Most of the bounds of convergence are usually defined with the symbol O. Giuliano Antonini and Weber (Bernoulli 23(4B):3268–3310, 2017) were the first who gave the explicit constant C of error bound $\frac{C}{\sqrt{n}}$ C n . In this paper, we improve the convergence rate and constants of error bounds in local limit theorem for $S_{n}$ S n . Our constants are less complicated than before, and thus easy to use.

Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights

We establish a quenched local central limit theorem for the dynamic random conductance model on $${\mathbb {Z}}^d$$ Z d only assuming ergodicity with respect to space-time 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.

Sharp Error Term in Local Limit Theorems and Mixing for Lorentz Gases with Infinite Horizon

We obtain sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This result allows us to further obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Hölder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case. We also obtain an asymptotic estimate for the tail probability of the first return time to the initial cell. In the process, we study families of transfer operators for infinite horizon Sinai billiards perturbed with the free flight function and obtain higher order expansions for the associated families of eigenvalues and eigenprojectors.

Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg–Landau $$\nabla \phi $$ Interface Model

We study a continuous-time random walk on $${\mathbb {Z}}^d$$ Z d in an environment of random conductances taking values in $$(0,\infty )$$ ( 0 , ∞ ) . For a static environment, we extend the quenched local limit theorem to the case of a general speed measure, given suitable ergodicity and moment conditions on the conductances and on the speed measure. Under stronger moment conditions, an annealed local limit theorem is also derived. Furthermore, an annealed local limit theorem is exhibited in the case of time-dependent conductances, under analogous moment and ergodicity assumptions. This dynamic local limit theorem is then applied to prove a scaling limit result for the space-time covariances in the Ginzburg–Landau $$\nabla \phi $$ ∇ ϕ model. We also show that the associated Gibbs distribution scales to a Gaussian free field. These results apply to convex potentials for which the second derivative may be unbounded.