Small time asymptotics of spectral heat content of isotropic processes
Rohan Sarkar
TL;DR
This work establishes a unifying small-time asymptotic for the spectral heat content $Q^X_\Omega(t)$ of translation-invariant isotropic processes $X$ in bounded domains with $C^{1,1}$ boundary, showing $\mathrm{Vol}(\Omega)-Q^X_\Omega(t) \sim \mu(t)\,\mathrm{Per}(\Omega)$ as $t\to 0$, where $\mu(t)=\mathbb{E}_0[\sup_{0\le s\le t} X^{(1)}_s\wedge 1]$. The paper develops a minimum-functional framework $Q^X_f(t)$ and leverages level-set representations to derive both lower and upper bounds, extending the result to $C^{1,κ}$ boundaries and a broad class of processes, including isotropic Lévy and Gaussian processes, as well as time-changed Brownian motions. For specific time-changes, explicit normalizations appear: inverse subordinators yield a $\sqrt{\varphi(1/t)}$-scaled limit with a universal constant, while self-similar time-changes give a power-law rate via $m(t)$. Together, these results connect small-time SHC to geometric perimeter, enabling analysis of non-Markovian models and broad process classes in probabilistic and PDE contexts.
Abstract
We provide a general approach for proving small time asymptotic of spectral heat content for any translation invariant isotropic process satisfying negligible tail probability condition. As a consequence, we recover several existing results in the context of Lévy processes and Gaussian processes, and provide spectral heat content asymptotic for a class of time-changed Brownian motions, which are non-Markovian.