Shiping Cao

activity

20162026

most citedExistence and uniqueness of diffusions on the Julia sets of Misiurewicz-Sierpinski maps

4 citations · 20 across the 27 of their papers we have counts for

collaborators

27 papers

math.PR2026

Homogenization of anisotropic diffusion on pre-Sierpiński carpets

Shiping Cao, Zhen-Qing Chen, Hua Qiu

We investigate the restoration of isotropy for anisotropic diffusions on pre-Sierpiński carpets in the plane, a problem previously studied by Barlow, Hattori, Hattori and Watanabe…

math.DS2026

Self-similar Dirichlet form on pillow-type carpets--a short analytic construction

Shiping Cao, Hua Qiu, Yizhou Wang

We give a short, self-contained analytic proof of the existence of self-similar Dirichlet forms on pillow-type carpets, a family of infinitely ramified fractals that includes the S…

math.PR2025

A non-canonical diffusion on the Sierpiński carpet

Shiping Cao, Hua Qiu, Bingshen Wang

We constructed a diffusion process on the Sierpiński carpet that satisfies the sub-Gaussian heat kernel estimate with respect to the Euclidean metric and a non-standard self-simila…

math.FA2024

Uniqueness and convergence of resistance forms on unconstrained Sierpinski carpets

Shiping Cao, Hua Qiu

We prove the uniqueness of self-similar $D_{4}$ -symmetric resistance forms on unconstrained Sierpinski carpets ( $U SC$ 's). Moreover, on a sequence of $U SC$ 's $K_n,…

math.FA2023

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

Shiping Cao, Zhen-Qing Chen, Takashi Kumagai

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 $\mathc…

math.FA2023

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

Shiping Cao, Hua Qiu

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 $R^{3}$ . I…

math.DS2022

Self-similar Dirichlet forms on polygon carpets

Shiping Cao, Hua Qiu, Yizhou Wang

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 carp…

math.PR2022

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

Shiping Cao

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…

math.FA2021

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

Shiping Cao, Qingsong Gu, Hua Qiu

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…

math.FA2021

A Sierpinski carpet like fractal without standard self-similar energy

Shiping Cao, Hua Qiu

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 ex…

math.AP2021★ 1 cited

Eigenvalues of Laplacians on Higher Dimensional Vicsek Set Graphs

Shiping Cao, Robert S. Strichartz, Melissa Wei

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, fi…

math.FA2021★ 2 cited

Dirichlet forms on unconstrained Sierpinski carpets

Shiping Cao, Hua Qiu

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 liv…

math.FA2020

Convergence of energy forms on Sierpinski gaskets with added rotated triangle

Shiping Cao

We study the convergence of resistance metrics and resistance forms on a converging sequence of spaces. As an application, we study the existence and uniqueness of self-similar Dir…

math.FA2020★ 1 cited

Brownian motion on the golden ratio Sierpinski gasket

Shiping Cao, Hua Qiu

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without any finitely ramified cell structure, via a study on…

math.FA2020★ 4 cited

Existence and uniqueness of diffusions on the Julia sets of Misiurewicz-Sierpinski maps

Shiping Cao, Malte S. Hassler, Hua Qiu +2

We study the balanced resistance forms on the Julia sets of Misiurewicz-Sierpinski maps, which are self-similar resistance forms with equal weights. In particular, we use a theorem…

math.FA2020★ 3 cited

Equivalence of Besov spaces on p.c.f. self-similar sets

Shiping Cao, Hua Qiu

On p.c.f. self-similar sets, of which the walk dimensions of heat kernels are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spa…

math.FA2020

Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems

Shiping Cao, Hua Qiu

We study the Sobolev spaces $H^{σ} (K)$ and $H_{0}^{σ} (K)$ on p.c.f. self-similar sets in terms of the boundary behavior of functions. First, for $σ \in R^{+}$ , we make an exact des…

math.FA2019

Spectral analysis beyond $ℓ^{2}$ on Sierpinski lattices

Shiping Cao, Yiqi Huang, Hua Qiu +2

We study the spectrum of the Laplacian on the Sierpinski lattices. First, we show that the spectrum of the Laplacian, as a subset of $C$ , remains the same for any $ℓ^{p}$ …

math.CA2019

A Trace Theorem For Sobolev Spaces On The Sierpinski Gasket

Shiping Cao, Shuangping Li, Robert S. Strichartz +1

We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the L2 domain of the Laplacian as a special…

math.CA2019

The Mathieu Differential Equation and Generalizations to Infinite Fractafolds

Shiping Cao, Anthony Coniglio, Xueyan Niu +2

One of the well-studied equations in the theory of ODEs is the Mathieu differential equation. A common approach for obtaining solutions is to seek solutions via Fourier series by c…

math.FA2019

Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions

Shiping Cao, Hua Qiu

We consider the Sobolev type spaces $H^{σ} (K)$ with $σ \geq 0$ , where $K$ is a post-critically finite self-similar set with the natural boundary. Firstly, we compare different classes…

math.FA2019★ 2 cited

Sobolev spaces and trace theorem on the Sierpinski gasket

Shiping Cao, Hua Qiu

On the Sierpinski gasket $S G$ , we consider Sobolev spaces $L_{σ}^{2} (S G)$ associated with the standard Laplacian $Δ$ with order $σ \geq 0$ . When $σ \in Z^{+}$ ,…

math.FA2018

Dirichlet forms on self-similar sets with overlaps

Shiping Cao, Hua Qiu

We study Dirichlet forms and Laplacians on self-similar sets with overlaps. A notion of "finitely ramified of finite type( $f . r . f . t .$ ) nested structure" for self-similar sets is int…

math.CA2017★ 1 cited

A topological proof of the non-degeneracy of harmonic structures on Sierpinski Gaskets

Shiping Cao, Hua Qiu

We present a direct proof of the non-degeneracy of the harmonic structures on the level- $n$ Sierpinski gaskets for any $n \geq 2$ , which was conjectured by Hino in [H1,H2] and confi…

Homogenization of anisotropic diffusion on pre-Sierpiński carpets

Self-similar Dirichlet form on pillow-type carpets--a short analytic construction

A non-canonical diffusion on the Sierpiński carpet

Uniqueness and convergence of resistance forms on unconstrained Sierpinski carpets

On Kigami's conjecture of the embedding Wp(K)⊂C(K)

Dirichlet forms on unconstrained Sierpinski carpets in R3

Self-similar Dirichlet forms on polygon carpets

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

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

A Sierpinski carpet like fractal without standard self-similar energy

Eigenvalues of Laplacians on Higher Dimensional Vicsek Set Graphs

Dirichlet forms on unconstrained Sierpinski carpets

Convergence of energy forms on Sierpinski gaskets with added rotated triangle

Brownian motion on the golden ratio Sierpinski gasket

Existence and uniqueness of diffusions on the Julia sets of Misiurewicz-Sierpinski maps

Equivalence of Besov spaces on p.c.f. self-similar sets

Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems

Spectral analysis beyond ℓ2 on Sierpinski lattices

A Trace Theorem For Sobolev Spaces On The Sierpinski Gasket

The Mathieu Differential Equation and Generalizations to Infinite Fractafolds

Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions

Sobolev spaces and trace theorem on the Sierpinski gasket

Dirichlet forms on self-similar sets with overlaps

A topological proof of the non-degeneracy of harmonic structures on Sierpinski Gaskets

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

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

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

Spectral analysis beyond $ℓ^{2}$ on Sierpinski lattices