Spectral Heat Content for Time-Changed Killed Brownian Motions

Abstract

The spectral heat content is investigated for time-changed killed Brownian motions on $$C^{1,1}$$ 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 in small time 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 $$\beta \in (\frac{1}{2}, 1)$$ .