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A unified approach to the small-time behavior of the spectral heat content for isotropic Lévy processes

Kei Kobayashi, Hyunchul Park

Abstract

This paper establishes the precise small-time asymptotic behavior of the spectral heat content for isotropic Lévy processes on bounded $C^{1,1}$ open sets of $\mathbb{R}^{d}$ with $d\ge 2$, where the underlying characteristic exponents are regularly varying at infinity with index $α\in (1,2]$, including the case $α=2$. Moreover, this asymptotic behavior is shown to be stable under an integrable perturbation of its Lévy measure. These results cover a wide class of isotropic Lévy processes, including Brownian motions, stable processes, and jump diffusions, and the proofs provide a unified approach to the asymptotic behavior of the spectral heat content for all of these processes.

A unified approach to the small-time behavior of the spectral heat content for isotropic Lévy processes

Abstract

This paper establishes the precise small-time asymptotic behavior of the spectral heat content for isotropic Lévy processes on bounded open sets of with , where the underlying characteristic exponents are regularly varying at infinity with index , including the case . Moreover, this asymptotic behavior is shown to be stable under an integrable perturbation of its Lévy measure. These results cover a wide class of isotropic Lévy processes, including Brownian motions, stable processes, and jump diffusions, and the proofs provide a unified approach to the asymptotic behavior of the spectral heat content for all of these processes.
Paper Structure (4 sections, 8 theorems, 82 equations)

This paper contains 4 sections, 8 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Examples

Key Result

Theorem 3.1

Let $D$ be a bounded $C^{1,1}$ open set in ${\mathbb R}^{d}$ with $d\ge 2$. Let $X$ be an isotropic Lévy process in ${\mathbb R}^d$ with the characteristic exponent $\psi\in {\mathcal{R}}_{\alpha}(\infty)$, $\alpha\in (1,2]$. Then where $\overline{Y}^{(\alpha)}$ is the supremum process of a symmetric $\alpha$-stable process $Y^{(\alpha)}$ in ${\mathbb R}$.

Theorems & Definitions (11)

  • Theorem 3.1
  • Corollary 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • Example 4.1: Pure Jump Process
  • Example 4.2: Jump Diffusion
  • ...and 1 more