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A non-canonical diffusion on the Sierpiński carpet

Shiping Cao, Hua Qiu, Bingshen Wang

TL;DR

The paper constructs a diffusion on the Sierpiński carpet driven by a non-standard, yet symmetric, self-similar measure and proves two-sided sub-Gaussian heat kernel estimates with respect to the Euclidean metric. Central to the approach are Knight move and corner move techniques adapted to heterogeneous cell weights, a Harnack inequality, and a detailed resistance/energy framework that yields an exponential scaling R_n ≍ λ^n. By scaling the pre-carpet diffusion and taking limits, the authors obtain a scaling-limit diffusion on the carpet with sharp heat kernel bounds, extending diffusion theory beyond symmetric p.c.f. fractals. The results broaden the class of fractal diffusions for which sub-Gaussian estimates hold and demonstrate robustness of the Knight move method under weight heterogeneity.

Abstract

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-similar measure.

A non-canonical diffusion on the Sierpiński carpet

TL;DR

The paper constructs a diffusion on the Sierpiński carpet driven by a non-standard, yet symmetric, self-similar measure and proves two-sided sub-Gaussian heat kernel estimates with respect to the Euclidean metric. Central to the approach are Knight move and corner move techniques adapted to heterogeneous cell weights, a Harnack inequality, and a detailed resistance/energy framework that yields an exponential scaling R_n ≍ λ^n. By scaling the pre-carpet diffusion and taking limits, the authors obtain a scaling-limit diffusion on the carpet with sharp heat kernel bounds, extending diffusion theory beyond symmetric p.c.f. fractals. The results broaden the class of fractal diffusions for which sub-Gaussian estimates hold and demonstrate robustness of the Knight move method under weight heterogeneity.

Abstract

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-similar measure.

Paper Structure

This paper contains 11 sections, 22 theorems, 177 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Knight moves
  4. Corner moves
  5. Knight moves
  6. Harnack Inequality
  7. Resistance Estimates
  8. Resistance and flows on $F_0^n$
  9. Upper bound estimate of $R_n$
  10. Lower bound estimate of $R_n$
  11. Scaling limits

Key Result

Theorem 1.1

There is a Feller process $((X_t)_{t>0},(P^x)_{x\in F})$ on the Sierpiński carpet $F$ with a transition density $q_t(x,y)$ satisfying the two-sided sub-Gaussian heat kernel estimate for all $x,y\in F$ and $t\leq 1$.

Figures (4)

  • Figure 1: The standard Sierpiński gasket and carpet.
  • Figure 2: Self-similar weight of $\mu$ with $\rho>0$.
  • Figure 3: $D_m(x)$ for $x\in \tilde{F}_0$.
  • Figure 4: $D_m(x)$ and $G_m(x)$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • ...and 34 more