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Brownian motion on the golden ratio Sierpinski gasket

Shiping Cao, Hua Qiu

Abstract

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 the trace of electrical networks on an infinite graph. The Dirichlet form is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. A theorem of uniqueness is also provided. Lastly, the associated process satisfies the two-sided sub-Gaussian heat kernel estimate.

Brownian motion on the golden ratio Sierpinski gasket

Abstract

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 the trace of electrical networks on an infinite graph. The Dirichlet form is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. A theorem of uniqueness is also provided. Lastly, the associated process satisfies the two-sided sub-Gaussian heat kernel estimate.

Paper Structure

This paper contains 9 sections, 24 theorems, 80 equations, 4 figures.

Table of Contents

  1. introduction
  2. Preliminary
  3. Resistance forms on the infinite graph $V_1$
  4. A renormalization map
  5. A fixed point problem
  6. The existence of a symmetric solution
  7. The uniqueness
  8. Construction of the Dirichlet form on $\mathcal{G}$
  9. Transition density estimate

Key Result

Theorem 1

There exists a unique strongly local regular resistance form $(\mathcal{E},\mathcal{F})$ on $\mathcal{G}$ such that $f\in\mathcal{F}$ if and only if $f\circ F_w\in \mathcal{F}$ for all $w\in W_1$ and $\sum_{w\in W_1} \rho_w^{-\theta}\mathcal{E}(f\circ F_w,f\circ F_w)<\infty$, where $\rho_w$ is the s Moreover, the form is decimation invariant with respect to the graph-directed construction of $\mat

Figures (4)

  • Figure 1: The golden ratio Sierpinski gasket $\mathcal{G}$.
  • Figure 2: A gasket with $0<\rho<1$ being a root of $x^4-2x+1=0$.
  • Figure 3: A graph-directed construction of $\mathcal{G}$.
  • Figure 4: The infinite graph $(V_1,\sim)$. (The bottom line equals to $\bar{V}_1\setminus V_1$.)

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2
  • Definition \oldthetheorem: finite type property
  • Definition \oldthetheorem: a graph-directed construction of $\mathcal{G}$
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 40 more