Related papers: Metrics on the Sierpinski carpet by weight functio…
We constructed a diffusion process on the Sierpi\'nski carpet that satisfies the sub-Gaussian heat kernel estimate with respect to the Euclidean metric and a non-standard self-similar measure.
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…
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…
We prove equivalent conditions for two-sided sub-Gaussian estimates of heat kernels on metric measure spaces.
We prove the existence of a strongly local, regular, self-similar Dirichlet form with a sub-Gaussian heat kernel estimate on an unconstrained Sierpinski carpet in $\mathbb{R}^3$. In the setting under consideration, the walk dimension $d_W$…
We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math. 1981) states that uniform domains in…
We prove that sub-Gaussian heat kernel estimates are inherited from a diffusion process on the ambient space to the reflected diffusion process on a subset which is an inner uniform domain.
We prove existence of a measurable Riemannian structure on higher-dimensional harmonic Sierpinski gasket fractals and deduce Gaussian heat kernel bounds in the geodesic metric. Our proof differs from that given by Kigami for the usual…
The relative configurational entropy per cell as a function of length scale is a sensitive detector of spatial self-similarity. For Sierpinski carpets the equally separated peaks of the above function appear at the length scales that depend…
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…
We introduce a H\"older regularity condition for harmonic functions on metric measure spaces and prove that, under a slow volume regular condition and an upper heat kernel estimate, the H\"older regularity condition, the weak Bakry-\'Emery…
We estimate the heat kernel on a closed Riemannian manifold $M$, with $dim(M)\geq 3$, evolving under the Ricci-harmonic map flow and the result depends on some constants arising from a Sobolev imbedding theorem. In a special case, when the…
In 2005, Liu et al. calculated the dimensionality of the intersection of Sierpinski carpet and a straight line with rational slope in the sense of Lebesgue measure.Sierpinski carpet is a self-similar set in two-dimensional planes obtained…
Given suitable functions $V, \Psi:[0,\infty) \to [0,\infty)$, we obtain necessary and sufficient conditions on $V,\Psi$ for the existence of a metric measure space and a symmetric diffusion process that satisfies sub-Gaussian heat kernel…
The computation of the distance between any two points of the Sierpinski Gasket with respect to the intrinsic metric has already been investigated by several authors. In the literature there is not an explicit formula using the code space…
This research is motivated by the study of the geometry of fractal sets and is focused on uniformization problems: transformation of sets to canonical sets, using maps that preserve the geometry in some sense. More specifically, the main…
This thesis is about local and non-local Dirichlet forms on the Sierpi\'nski gasket and the Sierpi\'nski carpet. We are concerned with the following three problems in analysis on the Sierpi\'nski gasket and the Sierpi\'nski carpet. First, a…
A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincar\'e inequalities. Our…
It is a long-standing open question to determine whether the Sierpi\'nski carpet attains its conformal dimension or not. While this problem remains unresolved, we prove that Cartesian products $\mathbb{S}^k$, where $\mathbb{S}$ is the…
We study the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat…