Higher order terms of the spectral heat content for killed subordinate and subordinate killed Brownian motions related to symmetric {\alpha}-stable processes in R

TL;DR

This paper analyzes the third term of the spectral heat content for killed subordinate Brownian motions related to symmetric alpha-stable processes, revealing the dependence on the interval length in a bounded domain.

Contribution

It provides a detailed analysis of the third term of spectral heat content for stable subordinators with specific indices, highlighting the role of the domain length.

Findings

The third term of spectral heat content depends on the length of the interval.

The analysis applies to stable subordinators with indices in (1, 2) and at 1.

The results extend understanding of spectral properties of subordinate Brownian motions.

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 \alpha is in (1, 2) or \alpha = 1. We prove that in the 3rd term of spectral heat content, one can observe the length b-a of the interval D.