Small-time heat decay for stable processes on fractal drums

TL;DR

This paper investigates how the spectral heat content decays over time for isotropic stable processes on fractal drums, extending previous work on subordinate killed Brownian motions and highlighting differences based on boundary complexity.

Contribution

It extends spectral heat content analysis from subordinate Brownian motions to stable processes on fractal domains under specific geometric conditions.

Findings

Decay rate differs from subordinate killed Brownian motions when -.

Shows substantial differences in heat decay based on boundary Minkowski dimension.

Provides new insights into heat behavior on fractal geometries.

Abstract

In this paper, we study the spectral heat content for isotropic stable processes on fractal drums (namely, open sets with fractal boundaries). The spectral heat content for subordinate killed Brownian motions by stable subordinators was investigated in \cite{PX23}, and the present work serves as a natural extension of \cite{PX23} for the spectral heat content for stable processes. Under suitable geometric conditions on the underlying domains, we show that the decay rate of the spectral heat content for stable processes differs substantially from that for subordinate killed Brownian motions when $\alpha=d-\b$, where $\b$ is the interior Minkowski dimension of the boundary of the underlying open set.