Higher Order Terms of the Spectral Heat Content for Killed Subordinate and Subordinate Killed Brownian Motions Related to Symmetric α-Stable Processes in ${\mathbb {R}}$

Abstract

We investigate the 3rd term of spectral heat content for killed subordinate and subordinate killed Brownian motions on a bounded open interval D = (a, b) in a real line when the underlying subordinators are stable subordinators with index α ∈ (1, 2) or α = 1. We prove that in the 3rd term of spectral heat content, one can observe the length b − a of the interval D.

Spectral heat content for α-stable processes in C1,1 open sets

“…The spectral heat content for isotropic stable processes was first studied in [2]. Since then, considerable progress has been made toward understanding the asymptotic behavior of the spectral heat content for other Lévy processes (see [1,7,10,12]).…”

“…We intend to deal with this topic in a future project. In the recent paper [10], a three-term asymptotic expansion of the spectral heat content of 1 dimensional symmetric α-stable processes, α ∈ [1, 2), was established. We believe that a similar result should hold true for d ≥ 2.…”

Large-time and small-time behaviors of the spectral heat content for time-changed stable processes

“…Recently, there have been growing interests in studying the spectral heat content for jump processes, which is defined by replacing the Brownian motions with other jump processes and putting the zero exterior condition outside Ω in order to take into account the fact that jump processes could exit the domain by jumping into R d \ Ω. In [1,2,4,6,8,9,10], the spectral heat content for stable processes and other Lévy processes are studied for both subordinate killed processes and killed subordinate processes. In particular, in [10], the two-term asymptotic expansion for the spectral heat content for isotropic stable processes on bounded C 1,1 open sets was investigated.…”

Large-time and small-time behaviors of the spectral heat content for time-changed stable processes

“…In [1,2,4,6,8,9,10], the spectral heat content for stable processes and other Lévy processes are studied for both subordinate killed processes and killed subordinate processes. In particular, in [10], the two-term asymptotic expansion for the spectral heat content for isotropic stable processes on bounded C 1,1 open sets was investigated.…”