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Local limit theorem of Brownian motion on metric trees

Soonki Hong

Abstract

Let $\mathcal{T}$ be a locally finite tree whose geometric boundary has infinitely many points. Suppose that a non-amenable group $\G$ acts isometrically and geometrically on the tree $\mathcal{T}$. In this paper, we show that if the length spectrum is Diophantine, then there exists a continuous function $C$ on $\mathcal{T}^2$ such that the heat kernel $p(t,x,y)$ of $\mathcal{T}$ satisfies $$\lim_{t\rightarrow \infty}t^{3/2}e^{λ_0t}p(t,x,y)=C(x,y)$$ for any $x,y\in \mathcal{T}$. Here, $λ_0$ is the bottom of the spectrum of the Laplacian on $\mathcal{T}$.

Local limit theorem of Brownian motion on metric trees

Abstract

Let be a locally finite tree whose geometric boundary has infinitely many points. Suppose that a non-amenable group acts isometrically and geometrically on the tree . In this paper, we show that if the length spectrum is Diophantine, then there exists a continuous function on such that the heat kernel of satisfies for any . Here, is the bottom of the spectrum of the Laplacian on .
Paper Structure (22 sections, 49 theorems, 259 equations, 8 figures)

This paper contains 22 sections, 49 theorems, 259 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The geometry of trees
  4. Laplacian and heat kernel on trees
  5. Properties of Green function
  6. Gibbs measure related to $\lambda$-Green functions
  7. Poincaré series and critical exponent of $f_\lambda$
  8. Patterson-Sullivan density of $f_\lambda$
  9. Gibbs measure of $f_\lambda$
  10. Variactional principle of Gibbs measure of $f_\lambda$
  11. Cross section of $\mathcal{G}\mathcal{T}$
  12. Coding of $\Gamma\backslash \mathcal{G}\mathcal{T}$
  13. Variational principle
  14. Uniform rapid mixing
  15. Uniform rapid mixing property of Gibbs measure of $f_\lambda$
  16. ...and 7 more sections

Key Result

Theorem 1.2

(Local Limit Theorem) Let $\mathcal{T}$ be a topologically complete locally finite metric tree. Every degree of a vertex is at least 3. Suppose that the cardinality of $\partial T$ is infinite and a discrete group $\Gamma$ acts isometrically and geometrically on $\mathcal{T}$. Suppose that the lengt

Figures (8)

  • Figure 1: Two geodesic lines $g_1$ and $g_2$ with $d_{\mathcal{G}\mathcal{T}}(g,g')<\varepsilon$
  • Figure 2: Strong Ancona inequality
  • Figure 3: $C(g,p,q)$
  • Figure 4: The construction of $\Theta$
  • Figure 5: The map $(g_-,g_+,t)\mapsto(w,g',s)\in \mathcal{G}_x^+\mathcal{T}\times W^+(w)\times \mathbb{R}$
  • ...and 3 more figures

Theorems & Definitions (90)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • ...and 80 more