Quenched Invariance Principle for the Random Walk on the Penrose Tiling

We consider the simple random walk on the graph corresponding to a Penrose tiling. We prove that the path distribution of the walk converges weakly to that of a non-degenerate Brownian motion for almost every Penrose tiling with respect to the appropriate invariant measure on the set of tilings. Our tool for this is the corrector method.Comment: 15 pages, 1 figur

Sub-Gaussian estimates of heat kernels on infinite graphs

We prove that a two sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay

Diffusive limits on the Penrose tiling

In this paper random walks on the Penrose lattice are investigated. Heat kernel estimates and the invariance principle are shown

Sub-Gaussian short time asymptotics for measure metric Dirichlet spaces

This paper presents estimates for the distribution of the exit time from balls and short time asymptotics for measure metric Dirichlet spaces. The estimates cover the classical Gaussian case, the sub-diffusive case which can be observed on particular fractals and further less regular cases as well. The proof is based on a new chaining argument and it is free of volume growth assumptions

From non-symmetric particle systems to non-linear PDEs on fractals

We present new results and challenges in obtaining hydrodynamic limits for non-symmetric (weakly asymmetric) particle systems (exclusion processes on pre-fractal graphs) converging to a non-linear heat equation. We discuss a joint density-current law of large numbers and a corresponding large deviations principle.Comment: v2: 10 pages, 1 figure. To appear in the proceedings for the 2016 conference "Stochastic Partial Differential Equations & Related Fields" in honor of Michael R\"ockner's 60th birthday, Bielefel

Random walk on the range of random walk

We study the random walk X on the range of a simple random walk on ℤ d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin

Hamiltonicity of random graphs produced by 2-processes

C1 - Refereed Journal Articl