Spectral heat content for time-changed killed Brownian motions

TL;DR

This paper studies the spectral heat content of time-changed killed Brownian motions in open sets, revealing how different types of time changes affect asymptotic behaviors depending on the Laplace exponent's index.

Contribution

It provides new asymptotic results for spectral heat content under various time-change mechanisms involving subordinators and inverse subordinators.

Findings

Asymptotic limit involves a probabilistic term depending on eta for inverse subordinators.

Universality in asymptotics holds for subordinators only when eta .5.

Different behaviors are observed depending on the type of subordinator used.

Abstract

The spectral heat content is investigated for time-changed killed Brownian motions on C1,1 open sets, where the time change is given by either a subordinator or an inverse subordinator, with the underlying Laplace exponent being regularly varying at \infty with index \beta \in (0, 1). In the case of inverse subordinators, the asymptotic limit of the spectral heat content is shown to involve a probabilistic term depending only on \beta \in (0, 1). In contrast, in the case of subordinators, this universality holds only when \beta \in ( 1/2 , 1).