Masanori Hino, Madoka Yasui
We study energy measures of canonical Dirichlet forms on inhomogeneous Sierpinski gaskets. We prove that the energy measures and suitable reference measures are mutually singular under mild assumptions.
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We determine the walk dimension of the Sierpiński gasket without using diffusion. We construct non-local regular Dirichlet forms on the Sierpiński gasket from regular Dirichlet forms on the Sierpiński graph whose suitable boundary is the Sierpiński gasket.
We prove that loop-erased random walks on finite pre-Sierpinski gaskets can be extended to the infinite pre-Sierpinski gasket by virtue of the `erasing-larger-loops-first' method, and obtain the asymptotic behavior of the walk as the number of steps increases, in particular, the displacement exponent and a law of the iterated logarithm.
We confirm, in a more general framework, a part of the conjecture posed by R. Bell, C.-W. Ho, and R. S. Strichartz [Energy measures of harmonic functions on the Sierpiński gasket, Indiana Univ. Math. J. 63 (2014), 831--868] on the distribution of energy measures for the canonical Dirichlet form on the two-dimensional standard Sierpinski gasket.
We obtain the Lifschitz tail asymptotics for the integrated density of states of the subordinate $α-$stable processes on the Sierpiński gasket $\mathcal G,$ evolving among killing Poissonian obstacles. Simultaneously, we derive the large-time asymptotics for the volume of the $α-$stable sausage on the gasket.
Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph approximations
本文在Sierpinski垫片上定义Dirichlet分数高斯场，证明其为离散分数高斯场的极限，建立了连续与离散场的联系。
Backward problems for stochastic differential equations on the Sierpinski gasket
本文研究谢尔宾斯基垫上随机微分方程的非线性倒向问题，证明了对应布朗鞅驱动的倒向随机微分方程解的存在唯一性，得到鞅加性泛函二次过程的指数可积性，并建立了半线性抛物型偏微分方程弱解的Feynman-Kac表示公式。
Standard resistance form on the infinite Sierpinski gasket and its trace onto the half-line $[0, \infty)$
本文构造无限谢尔宾斯基垫片逼近图上的相容电阻序列，证明其极限诱导对应布朗运动的狄利克雷形式，进而得到该布朗运动到半直线[0, ∞)的迹过程跳跃核的显式表达式。
Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph $G$, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph the asymptotic shape is known to be a ball in the usual graph metric. In this paper we establish bounds for the fluctuations of the cluster from its asymptotic shape.
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.
We consider a model of loop-erased random walks on the finite pre-Sierpinski gasket which permits rigorous analysis. We prove the existence of the scaling limit and show that the path of the limiting process is almost surely self-avoiding, while having Hausdorff dimension strictly greater than 1. This result means that the path has infinitely fine creases, while having no self-intersection. Our loop-erasing procedure is formulated by a `larger-scale-loops-first' rule. It enables us to obtain exact recursion relations, making use of `self-similarity' of a fractal structure.
We show that the `erasing-larger-loops-first' (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpinski gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpinski gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the `standard' self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The e…
Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover
本文证明了$l$级Sierpinski垫片上的典型扩散经重标度后收敛到初始三角形上的Brownian motion，并建立了转移密度的局部中心极限定理，核心是借助瓶颈间的欧几里得混合推导Poincaré不等式。
On the structure of the sandpile identity element on Sierpinski gasket graphs
研究Sierpinski垫片有限近似图的可交换沙堆群恒等元，证明其标度极限中的二阶项收敛到垫片上到最近角的路径距离。
We establish the Lifschitz-type singularity around the bottom of the spectrum for the integrated density of states for a class of subordinate Brownian motions in presence of the nonnegative Poissonian random potentials, possibly of infinite range, on the Sierpiński gasket. We also study the long-time behaviour for the corresponding averaged Feynman-Kac functionals.
We obtain the equilibrium fluctuations for the empirical density of particles for the zero-range process in the Sierpinski gasket. The limiting process is a generalized Ornstein-Uhlenbeck process generated by the Neumann Laplacian and its corresponding Dirichlet form on the gasket.
A stochastic Pontryagin maximum principle on the Sierpinski gasket
本文研究Sierpinski垫片上的随机控制问题，通过热核估计推导阶比较引理，结合BSDEs技术建立含两个必要性方程的随机庞特里亚金极大值原理，揭示分形空间测度奇异性的影响，并以线性调节器问题为例验证。
The divisible sandpile model is a growth model on graphs that was introduced by Levine and Peres as a tool to study internal diffusion limited aggregation. In this work we investigate the shape of the divisible sandpile model on the graphical Sierpinski gasket SG. We show that the shape is a ball in the graph metric of SG. Moreover we give an exact representation of the odometer function of the divisible sandpile.
The aim of the current work is to investigate structural properties of the sandpile group of a special class of self-similar graphs. More precisely, we consider Abelian sandpiles on Sierpinski gasket graphs and for the choice of normal boundary conditions, we give a characterization of the identity element and a recursive description of the sandpile group. Finally, we consider Abelian sandpile Markov chains on the aforementioned graphs and we improve the existing bounds on the speed of convergence to stationarity.
We prove that the hydrodynamic limit of a zero-range process evolving in graphs approximating the Sierpinski gasket is given by a nonlinear heat equation. We also prove existence and uniqueness of the hydrodynamic equation by considering a finite-difference scheme.
We consider the discrete time quantum random walks on a Sierpinski gasket. We study the hitting probability as the level of fractal goes to infinity in terms of their localization exponents $β_w$ , total variation exponents $δ_w$ and relative entropy exponents $η_w$ . We define and solve the amplitude Green functions recursively when the level of the fractal graph goes to infinity. We obtain exact recursive formulas for the amplitude Green functions, based on which the hitting probabilities and expectation of the first-passage time are calculated. Using the recursive formula with the aid of Mo…
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…
Estimates of the local spectral dimension of the Sierpinski gasket
本文受 Kusuoka 测度分布研究启发，对二维 Sierpinski 垫片的局部谱维数给出定量估计。
Brownian motion on the golden ratio Sierpinski gasket
本文针对无有限分支胞腔结构的黄金比例 Sierpinski 垫片，利用无限图电网络迹构造自相似、抽稀不变的强局部正则 Dirichlet 型，证明唯一性并得到相关过程的双侧次高斯热核估计。
Pointwise regularity and irregularity of energy densities on $N$-dimensional Sierpinski gaskets
研究$N$维Sierpinski垫片上调和函数关于Kusuoka测度的能量密度的逐点正则性，证明非常值调和函数的每个Borel代表元在一个满Kusuoka测度集上每点不连续，而在每个一维边上存在规范逐点版本且具有显式最优Hölder指数。
We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in th…
Internal diffusion-limited aggregation (IDLA) is a stochastic growth model on a graph $G$ which describes the formation of a random set of vertices growing from the origin (some fixed vertex) of $G$. Particles start at the origin and perform simple random walks; each particle moves until it lands on a site which was not previously visited by other particles. This random set of occupied sites in $G$ is called the IDLA cluster. In this paper we consider IDLA on Sierpinski gasket graphs, and show that the IDLA cluster fills balls (in the graph metric) with probability 1.
Heat kernel lower bound estimates for symmetric pure jump processes via averaged jump kernels
利用平均跳跃核，在一般体积加倍度量测度空间上推导对称纯跳过程的热核下界估计，并应用于Sierpinski垫片底部布朗运动迹的转移密度函数下界。
The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by unit complex-valued weights $\{ω_{xy}\}_{xy\in E}$, satisfying the condition that $ω_{xy}=\overline{ω_{yx}}$ for every directed edge $xy$. When properly interpreted, these complex weights give rise to magnetic fluxes through cycles in the graph. In this paper we establish the spectrum of the magnetic Laplacian, as a set of real numbers with multiplicities, on the Sierpinski gasket graph ($SG$) where the magnetic fluxes…
Internal aggregation models with multiple sources and obstacle problems on Sierpinski gaskets
本文研究 Sierpinski 垫片图上三种内部聚合模型（可整除沙堆、内部 DLA 和转子聚合）在多源初始构型下的缩放极限，证明其收敛到分形上自由边界问题的确定性解，并分析解的边界正则性。
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.
We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner vertex. In particular, the abelian sandpile growth problem is exactly solved via a recursive construction of self-similar sandpile tiles. We show that sandpile growth and patterns exhibit a $(2\cdot 3^n)$-periodicity as a function of the initial mass. Moreover, the cluster explodes---increments by more than 1 in radius---at periodic intervals, a phenomenon not seen on $\mathbb{Z}^d$ or trees. We explici…
Using the method of spectral decimation and a modified version of Kirchhoffs Matrix-Tree Theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in Theorem (3.4). Examples calculated include the Sierpinski Gasket, a non p.c.f. analog of the Sierpinski Gasket, the Diamond fractal, and the Hexagasket. For each example, the asymptotic complexity constant is found. Dropping the fully symmetry assumption, it is shown that the limsup and liminf of the asymptotic complexity constant exi…
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