Martin T. Barlow, Richard F. Bass, Takashi Kumagai, Alexander Teplyaev
We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.
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We prove that the critical probability for the Sierpinski carpet lattice in two dimensions is uniquely determined. The transition is sharp. This extends the Kumagai's result to the original Sierpinski carpet lattice.
We prove the Boundary Harnack Principle related to fractional powers of Laplacian for some natural regions in the two-dimensional Sierpinski carpet. This is a natual application of a probabilistic method based on the Ikeda-Watanabe formula
Uniform Harnack inequalities for harmonic functions on the pre- and graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the construction of Brownian motion on Sierpinski carpets embedded in $\R^d$, $d\geq 3$, estimates on the fundamental solution of the heat equation, and Sobolev and Poincaré inequalities.
We construct a Sierpinski carpet like fractal, on which a self-similar diffusion with sub-Gaussian heat kernel estimate does not exist, in contrast to previous researches on the existence of such diffusions, on the generalized Sierpinski carpets and recently introduced unconstrained Sierpinski carpets.
We give an elementary self-contained proof of the fact that the walk dimension of the Brownian motion on an arbitrary generalized Sierpiński carpet is greater than two, no proof of which in this generality had been available in the literature. Our proof is based solely on the self-similarity and hypercubic symmetry of the associated Dirichlet form and on several very basic pieces of functional analysis and the theory of regular symmetric Dirichlet forms. We also present an application of this fact to the singularity of the energy measures with respect to the canonical self-similar measure (uni…
Dirichlet forms on unconstrained Sierpinski carpets
本文在无约束Sierpinski地毯上构造对称自相似Dirichlet形式，推广了平面情形并放宽几何与对角假设。
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。
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。
Consider the free field on a fractal graph based on a high-dimensional Sierpinski carpet (e.g. the Menger sponge), that is, a centered Gaussian field whose covariance is the Green's function for simple random walk on the graph. Moreover assume that a "hard wall" is placed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on $\mathbb{Z}^d$, $d \geq 3$, to the …
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。
We prove the existence of the integrated density of states for subordinate Brownian motions in presence of the Poissonian random potentials on the Sierpiński gasket.
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…
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.
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。
In this paper we identify the domain of the Dirichlet form associated with the Brownian motion on simple nested fractals with an integral Lipschitz space. This result generalizes such an identification on the Sierpiński gasket, carried on by Jonsson.
We derive the modulus of continuity of a class of processes called Canonical Embedded Branching Processes (CEBP), recently introduced by Decrouez and Jones, and we establish their monofractal character. CEBP provide a rich class of processes, including the Brownian motion as a particular case. The techniques developed in this study follow the steps of Barlow and Perkins on Brownian motion on a Sierpinski gasket, though complications arise here since CEBP are not Markovian in general.
Let $X$ be a fractional Brownian motion in $\mathbb{R}^d$. For any Borel function $f:[0,1] \to \mathbb{R}^d$, we express the Hausdorff dimension of the image and the graph of $X+f$ in terms of $f$. This is new even for the case of Brownian motion and continuous $f$, where it was known that this dimension is almost surely constant. The expression involves an adaptation of the parabolic dimension previously used by Taylor and Watson to characterize polarity for the heat equation. In the case when the graph of $f$ is a self-affine McMullen-Bedford carpet, we obtain an explicit formula for the dim…
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