Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

TL;DR

This paper investigates the small-time asymptotic behavior of spectral heat content for fractal boundary sets in Euclidean space, driven by subordinate killed Brownian motions with stable subordinators, revealing dependence on geometric and arithmetic properties.

Contribution

It extends spectral heat content analysis to fractal sets with subordinate Brownian motions, highlighting the influence of boundary Minkowski dimension and arithmetic properties on asymptotics.

Findings

Asymptotic behavior depends on whether the logarithms of similitude coefficients are arithmetic.

Results cover the full range of stability index $\alpha ext{ in } (0,2)$.

Main tools include previous Brownian motion results and renewal theorem.

Abstract

We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in $\mathbb{R}^{d}$, $d\geq 1$, with respect to subordinate killed Brownian motions via $\alpha/2$-stable subordinators and establish the asymptotic behavior of the spectral heat content as $t\to 0$ for the full range of $\alpha\in (0,2)$. Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when $\alpha\in [d-\b,2)$, where $\b$ is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.