Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local Central Limit Theorem for Reflecting Diffusions in a Continuum Percolation Cluster

Yutaka Takeuchi

Abstract

Reflecting diffusions on continuum percolation clusters are considered. Assuming that the occupied region has a unique unbounded cluster and the cluster satisfies geometrical conditions such as volume regularity, isoperimetric conditions, and a hole size condition, we prove a quenched local central limit theorem for reflecting diffusions on the cluster.

Local Central Limit Theorem for Reflecting Diffusions in a Continuum Percolation Cluster

Abstract

Reflecting diffusions on continuum percolation clusters are considered. Assuming that the occupied region has a unique unbounded cluster and the cluster satisfies geometrical conditions such as volume regularity, isoperimetric conditions, and a hole size condition, we prove a quenched local central limit theorem for reflecting diffusions on the cluster.
Paper Structure (6 sections, 27 theorems, 175 equations)

This paper contains 6 sections, 27 theorems, 175 equations.

Table of Contents

  1. Introduction and result
  2. Parabolic Harnack inequality and Hölder continuity for a deterministic model
  3. Sobolev inequality and Poincaré inequality
  4. Cutoff inequalities
  5. Parabolic Harnack inequality and Hölder continuity
  6. Local central limit theorem and the proof of the main theorem

Key Result

Theorem 1.2

Let $d \geq 2$. Suppose that Assumptions asm:erg-asm:volIso hold. Let $P_0^\omega$ be the law of $\{X_t^\omega\}_t$ starting at $0$. Then for $\widehat{\mathbb{P}}$-a.s. $\omega$, the scaled process $\{\varepsilon X_{\varepsilon ^{-2}t}^\omega\}_t$ under $P_0^\omega$ converges in law to a Brownian m

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2: Quenched invariance principle, Y
  • Theorem 1.3: Local central limit theorem
  • Example 1.4: Reflecting Brownian motion on a Poisson Boolean model
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5: Local Sobolev inequality
  • ...and 38 more