An elementary proof that walk dimension is greater than two for Brownian motion on Sierpiński carpets

$\u211d^3$中无约束Sierpinski地毯上的Dirichlet型

Dirichlet forms on unconstrained Sierpinski carpets in $\mathbb{R}^3$

该研究证明三维无约束Sierpinski地毯上存在带次高斯热核估计的强局部正则自相似Dirichlet型，且其Hausdorff维数恒大于行走维数。

无约束Sierpinski地毯上阻力形式的唯一性与收敛性

Uniqueness and convergence of resistance forms on unconstrained Sierpinski carpets

本文证明了无约束Sierpinski地毯上自相似$D_4$对称阻力形式的唯一性，并揭示了地毯序列在Hausdorff度量下收敛时，关联扩散过程依分布收敛与测地度量等度连续的等价条件。

Sierpiński 地毯上 $p$-能量及相关能量测度的构造

Construction of $p$-energy and associated energy measures on Sierpiński carpets

本文在广义 Sierpiński 地毯上构造离散 $p$-能量的标度极限及相关能量测度，并刻画相应函数空间与 $p=2$ 时的经典 Dirichlet 型。

Fractional Gaussian fields on the Sierpinski gasket and related fractals

We define and study a fractional Gaussian field $X$ with Hurst parameter $H$ on the Sierpiński gasket $K$ equipped with its Hausdorff measure $μ$. It appears as a solution, in a weak sense, of the equation $(-Δ)^s X =W$ where $W$ is a Gaussian white noise on $L_0^2(K,μ)$, $Δ$ the Laplacian on $K$ and $s= \frac{d_h+2H}{2d_w}$, where $d_h$ is the Hausdorff dimension of $K$ and $d_w$ its walk dimension. The construction of those fields is then extended to other fractals including the Sierpiński carpet.

当 $κ$ 接近 8 时，Schramm-Loewner evolution 包含一个拓扑 Sierpiński carpet

Schramm-Loewner evolution contains a topological Sierpiński carpet when $κ$ is close to 8

本文证明了当 $κ$ 接近 8 时，SLE$_κ$ 曲线的值域几乎必然包含拓扑 Sierpiński carpet，从而得出 SLE$_κ$ 共形不可移除且共形焊接问题无唯一解，并证明其互补连通分量的邻接图不连通。

Sierpiński地毯上的一阶Sobolev空间、自相似能量与能量测度

First-order Sobolev spaces, self-similar energies and energy measures on the Sierpiński carpet

本文在平面Sierpiński地毯上对所有p构造一阶Sobolev空间、p-能量及能量测度，利用Loewner型估计证明Poincaré不等式，并应用于Ahlfors正则共形维数的可达性。

Self-similar Dirichlet forms on polygon carpets

We construct symmetric self-similar diffusions with sub-Gaussian heat kernel estimates on two types of polygon carpets, which are natural generalizations of planner Sierpinski carpets (SC). The first ones are called perfect polygon carpets that are natural analogs of SC in that any intersection cells are either side-to-side or point-to-point. The second ones are called bordered polygon carpets which satisfy the boundary including condition as SC but allow distinct contraction ratios.

A trace theorem for Dirichlet forms on fractals

We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets to their boundaries, where boundaries mean the triangles and rectangles which confine gaskets and carpets. As an application, we construct diffusion processes on a collection of fractals called fractal fields, which behave as the appropriate fractal diffusion within each fractal component of the field.

Densely defined non-closable curl on carpet-like metric measure spaces

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have positive two-dimensional Lebesgue measure and carry nontrivial $2$-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on $1$-forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abst…

Boundary Harnack Inequality for alpha-harmonic functions on the Sierpiński triangle

We prove an uniform boundary Harnack inequality for nonnegative functions harmonic with respect to $α$-stable process on the Sierpi{ń}ski triangle, where $α\in (0, 1)$. Our result requires no regularity assumptions on the domain of harmonicity.

Upper estimate of martingale dimension for self-similar fractals

We study upper estimates of the martingale dimension $d_m$ of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that $d_m=1$ for natural diffusions on post-critically finite self-similar sets and that $d_m$ is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.

暂态图上带杀伤的随机游走的热核估计

Heat Kernel Estimates for Random Walks with Killing on Transient Graphs

针对暂态图上带杀伤的随机游走，研究人员对加权图上薛定谔算子Δ+W的热核给出匹配的双侧估计，通过两种方法证明结果，适用于分形型图等示例。

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space $X$, our paper studies elements of vector analysis, $L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on $X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets.

The box dimension of random box-like self-affine sets

In this paper we study two random analogues of the box-like self-affine attractors introduced by Fraser, itself an extension of Sierpiński carpets. We determine the almost sure box-counting dimension for the homogeneous random case ($1$-variable random), and give a sufficient condition for the almost sure box dimension to be the expectation of the box dimensions of the deterministic attractors. Furthermore we find the almost sure box-counting dimension of the random recursive model ($\infty$-variable), which includes affine fractal percolation.