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

TL;DR

This paper investigates the asymptotic behaviors of spectral heat content for time-changed stable processes, revealing polynomial decay at large times and multiple regimes at small times influenced by the subordinator's Laplace exponent.

Contribution

It provides a comprehensive analysis of spectral heat content decay for a broad class of inverse subordinators, contrasting with classical exponential decay results.

Findings

Spectral heat content decays polynomially at large times with a rate linked to the subordinator's Laplace exponent.

At small times, the decay exhibits three regimes influenced by both the Laplace exponent and the stable process index.

Large-time behavior differs significantly from the case with standard subordinators, showing polynomial rather than exponential decay.

Abstract

We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.