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

2023/06/20 by Kobayashi, Kei, Park, Hyunchul
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2306.11690

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 ℝ^d with d≥ 2, where the underlying characteristic exponents are regularly varying at infinity with index α∈ (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.

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• Small time asymptotics of spectral heat content of isotropic processes
• Dichotomy in the small-time asymptotics of spectral heat content for Lévy processes
• Asymptotic expansion of the nonlocal heat content
• Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic α-stable processes
• Spectral heat content for time-changed killed Brownian motions