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Small-time heat decay for stable processes on fractal drums

Hyunchul Park, Yimin Xiao

Abstract

In this paper, we study the spectral heat content for isotropic stable processes on fractal drums (namely, open sets with fractal boundaries). The spectral heat content for subordinate killed Brownian motions by stable subordinators was investigated in \cite{PX23}, and the present work serves as a natural extension of \cite{PX23} for the spectral heat content for stable processes. Under suitable geometric conditions on the underlying domains, we show that the decay rate of the spectral heat content for stable processes differs substantially from that for subordinate killed Brownian motions when $α=d-\b$, where $\b$ is the interior Minkowski dimension of the boundary of the underlying open set.

Small-time heat decay for stable processes on fractal drums

Abstract

In this paper, we study the spectral heat content for isotropic stable processes on fractal drums (namely, open sets with fractal boundaries). The spectral heat content for subordinate killed Brownian motions by stable subordinators was investigated in \cite{PX23}, and the present work serves as a natural extension of \cite{PX23} for the spectral heat content for stable processes. Under suitable geometric conditions on the underlying domains, we show that the decay rate of the spectral heat content for stable processes differs substantially from that for subordinate killed Brownian motions when , where is the interior Minkowski dimension of the boundary of the underlying open set.
Paper Structure (4 sections, 9 theorems, 63 equations)

This paper contains 4 sections, 9 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Case Studies: the Complement of the Ternary Cantor Set and the modified Sierpiński Gasket

Key Result

Theorem 2.1

Suppose that a map $f:{\mathbb R}\to {\mathbb R}$ satisfies the renewal equation eqn:RE with and for some constants $c_1,c_2>0$. Then, the solution of the renewal equation eqn:RE is given by Furthermore, if $\{\gamma_{j}\}$ is non-arithmetic, then If $\{\gamma_{j}\}$ is arithmetic with span $\gamma$, then

Theorems & Definitions (15)

  • Theorem 2.1: Renewal Theorem LV1996
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Remark 3.9
  • Lemma 3.10
  • Corollary 3.11
  • ...and 5 more