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Shiping Cao

Publications and source records attributed to Shiping Cao.

At least 19 recordsLinked to original sources
arXiv2026

Homogenization of anisotropic diffusion on pre-Sierpi\'{n}ski carpets

We investigate the restoration of isotropy for anisotropic diffusions on pre-Sierpi\'nski carpets in the plane, a problem previously studied by Barlow, Hattori, Hattori and Watanabe \cite{BHHW} in which they obtained a weak homogenization property for the ratio of effective resistances of the anisotropic diffusions. We establish the weak convergence of these anisotropic diffusions to Brownian motion on the Sierpi\'nski carpet. As a consequence, we give an affirmative answer to the strong homogenization conjecture raised in \cite[p.3]{BHHW}.

math.PR↗
arXiv2026

Heat kernel for reflected jump diffusion on Ahlfors regular domains

We study reflected jump diffusions on Ahlfors regular domains in general metric measure spaces. Under the condition that the Dirichlet form on the ambient space satisfies a capacity upper bound estimate, we construct an extension operator from the reflected Dirichlet space to the ambient Dirichlet space, with a scale-invariant local bound. Second, we establish the mixed stable-like heat kernel estimates for the reflected jump diffusion, assuming that the process on the ambient space satisfies the same type of heat kernel estimates.

math.PR↗
arXiv2024

Uniform boundary Harnack principle for non-local operators on metric measure spaces

We obtain a uniform boundary Harnack principle (BHP) on any open sets for a large class of non-local operators on metric measure spaces under a jump measure comparability and tail estimate condition, and an upper bound condition on the distribution function for the exit times from balls. These conditions are satisfied by any non-local operator $\mathcal{L}$ that admits a two-sided mixed stable-like heat kernel bounds when the underlying metric measure spaces have volume doubling and reverse volume doubling properties. The results of this paper are new even for non-local operators on Euclidean spaces. In particular, our results give not only the scale invariant but also uniform BHP for the first time for non-local operators on Euclidean spaces of both divergence form and non-divergence form with measurable coefficients.

math.PR↗
arXiv2024

Boundary trace theorems for symmetric reflected diffusions

Starting with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$, one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the theory of reflected Dirichlet spaces. The boundary trace process $\check X$ of $X$ on the boundary $\partial D:=D^*\setminus D$ is the reflected diffusion process $\bar X$ time-changed by a smooth measure $\nu$ having full quasi-support on $\partial D$. The Dirichlet form of the trace process $\check X$ is called the trace Dirichlet form. In the first part of the paper, we give a Besov space type characterization of the domain of the trace Dirichlet form for any good smooth measure $\nu$ on the boundary $\partial D$. In the second part of this paper, we study properties of the harmonic measure of $\bar X$ on the boundary $\partial D$. In particular, we provide a condition equivalent to the doubling property of the harmonic measure. Finally, we characterize and provide estimates of the jump kernel of the trace Dirichlet form under the doubling condition of the harmonic measure on $\partial D$.

math.PR↗
arXiv2024

Uniqueness and convergence of resistance forms on unconstrained Sierpinski carpets

We prove the uniqueness of self-similar $D_4$-symmetric resistance forms on unconstrained Sierpinski carpets ($\mathcal{USC}$'s). Moreover, on a sequence of $\mathcal{USC}$'s $K_n, n\geq 1$ converging in Hausdorff metric, we show that the associated diffusion processes converge in distribution if and only if the geodesic metrics on $K_n, n\geq 1$ are equicontinuous with respect to the Euclidean metric.

math.FA↗
arXiv2024

Whether $p$-conductive homogeneity holds depends on $p$

We introduce two fractals, in Euclidean spaces of dimension two and three respectively, such the $2$-conductive homogeneity holds but there is some $\eps \in (0, 1)$ so that the $p$-conductive homogeneity fails for every $p\in (1, 1+\eps)$. In addition, these two fractals have Ahlfors regular conformal dimension within the interval $(1, 2)$ and $(2, 3)$, respectively.

math.FA↗
arXiv2023

On Kigami's conjecture of the embedding $\mathcal{W}^p(K)\subset C(K)$

Let $(K,d)$ be a connected compact metric space and $p\in (1, \infty)$. Under the assumption of \cite[Assumption 2.15]{Ki2} and the conductive $p$-homogeneity, we show that $\mathcal{W}^p(K)\subset C(K)$ holds if and only if $p>\operatorname{dim}_{AR}(K,d)$, where $\mathcal{W}^p(K)$ is Kigami's $(1,p)$-Sobolev space and $\operatorname{dim}_{AR}(K,d)$ is the Ahlfors regular dimension.

math.FA↗
arXiv2023

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

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 $d_W$ and the Hausdorff dimension $d_H$ always satisfy the inequality that $d_H>d_W$.

math.FA↗
arXiv2022

Scaling limits of loop-erased Markov chains on resistance spaces via a partial loop-erasing procedure

We introduce partial loop-erasing operators. We show that by applying a refinement sequence of partial loop-erasing operators to a finite Markov chain, we get a process equivalent to the chronological loop-erased Markov chain. As an application, we construct loop-erased random paths on bounded domains of resistance spaces as the weak limit of the loop erasure of the Markov chains on a sequence of finite sets approximating the space, and the limit is independent of the approximating sequences. The random paths we constructed are simple paths almost surely, and they can be viewed as the loop-erasure of the paths of the diffusion process. Finally, we show that the scaling limit of the loop-erased random walks on the Sierpiński carpet graphs exists, and is equivalent to the loop-erased random paths on the Sierpińksi carpet.

math.PR↗
arXiv2022

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.

math.DS↗
arXiv2021

$p$-energies on p.c.f. self-similar sets

We study $p$-energies on post critically finite (p.c.f.) self-similar sets for $1<p<\infty$, as limits of discrete $p$-energies on approximation graphs, extending the construction of Dirichlet forms, the $p=2$ setting. By suitably enlarging the choices of discrete $p$-energies, and employing the energy averaging method developed by Kusuoka-Zhou, we prove the existence of symmetric $p$-energies on affine nested fractals, and extend Sabot's celebrated criterion for existence and non-existence of Dirichlet forms on p.c.f. self-similar sets to the $1<p<\infty$ setting.

math.FA↗
arXiv2021

A Sierpinski carpet like fractal without standard self-similar energy

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.

math.FA↗
arXiv2021

Spectral analysis beyond $\ell^2$ on Sierpinski lattices

We study the spectrum of the Laplacian on the Sierpinski lattices. First, we show that the spectrum of the Laplacian, as a subset of $\mathbb{C}$, remains the same for any $\ell^p$ spaces. Second, we characterize all the spectral points for the lattices with a boundary point.

math.FA↗
arXiv2021

Eigenvalues of Laplacians on Higher Dimensional Vicsek Set Graphs

We study the graphs associated with Vicsek sets in higher dimensional settings. First, we study the eigenvalues of the Laplacians on the approximating graphs of the Vicsek sets, finding a general spectral decimation function. This is an extension of earlier results on two dimensional Vicsek sets. Second, we study the Vicsek set lattices, which are natural analogues to the Sierpinski lattices. We have a criterion when two different Vicsek set lattices are isomorphic.

math.AP↗
arXiv2021

Dirichlet forms on unconstrained Sierpinski carpets

We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the $1/k$ grids. The intersection of two cells can be a line segment of irrational length, and the non-diagonal assumption is dropped in this recurrent setting.

math.FA↗