The random conductance model with heavy tails on nested fractal graphs

Reflected Brownian motion on simple nested fractals

We prove the existence of the reflected diffusion on a complex of an arbitrary size for a large class of planar simple nested fractals. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions on the fractal under which this projection can be well defined. They are illustrated by various specific examples. We first construct a proper version of the transition probability densities for reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman-Ko…

Discrete Fractal Dimensions of the Ranges of Random Walks in $\Z^d$ Associate with Random Conductances

Let X= {X_t, t \ge 0} be a continuous time random walk in an environment of i.i.d. random conductances {μ_e \in [1, \infty), e \in E_d}, where E_d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice Z^d and d\ge 3. Let R = {x \in Z^d: X_t = x for some t \ge 0} be the range of X. It is proved that, for almost every realization of the environment, dim_H (R) = dim_P (R) = 2 almost surely, where dim_H and dim_P denote respectively the discrete Hausdorff and packing dimension. Furthermore, given any set A \subseteq Z^d, a criterion for A to be hit by X_t for arbitrarily large …

Density of states for the Anderson model on nested fractals

We prove the existence and establish the Lifschitz singularity of the integrated density of states for certain random Hamiltonians $H^ω=H_0+V^ω$ on fractal spaces of infinite diameter. The kinetic term $H_0$ is given by $ϕ(-\mathcal L),$ where $\mathcal L$ is the Laplacian on the fractal and $ϕ$ is a completely monotone function satisfying some mild regularity conditions. The random potential $V^ω$ is of alloy-type.

嵌套分形上的 $L^p$-Poincaré 不等式

$L^p$-Poincaré inequalities on nested fractals

本文利用热核方法发展分形上 Korevaar-Schoen-Sobolev 空间的局部 $L^p$ 理论，证明了嵌套分形上度量球的尺度不变 $L^p$-Poincaré 不等式，并应用于 Sobolev 不等式、极大函数及 Hajłasz-Sobolev 空间研究。

在嵌套分形上带有泊松随机环境的从属布朗运动的IDS

IDS for subordinate Brownian motions in Poisson random environment on nested fractals

本文研究了在嵌套分形上带有泊松随机环境的从属布朗运动的IDS奇异性，通过将随机势能转化为合金型势能，拓展了伯恩斯坦函数的应用范围。

离散时间马尔可夫链及其在电网络上的局部时间的缩放极限

Scaling limits of discrete-time Markov chains and their local times on electrical networks

本文研究了离散时间马尔可夫链在电网络上的局部时间收敛性，通过分析扩展Dirichlet空间和电网络的迹，证明了在特定条件下序列的收敛性。

Spectral Asymptotics for $V$-variable Sierpinski Gaskets

The family of $V$-variable fractals provides a means of interpolating between two families of random fractals previously considered in the literature; scale irregular fractals ($V=1$) and random recursive fractals ($V=\infty$). We consider a class of $V$-variable affine nested fractals based on the Sierpinski gasket with a general class of measures. We calculate the spectral exponent for a general measure and find the spectral dimension for these fractals. We show that the spectral properties and on-diagonal heat kernel estimates for $V$-variable fractals are closer to those of scale irregular…

Local limit theorems for sequences of simple random walks on graphs

In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.

Relations between scaling exponents in unimodular random graphs

We investigate the validity of the "Einstein relations" in the general setting of unimodular random networks. These are equalities relating scaling exponents: $d_w = d_f + \tildeζ$ and $d_s = 2 d_f/d_w$, where $d_w$ is the walk dimension, $d_f$ is the fractal dimension, $d_s$ is the spectral dimension, and $\tildeζ$ is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if $d_f$ and $\tildeζ \geq 0$ exist, then $d_w$ and $d_s$ exist, and the aforementioned …