Asymptotic expansion of the nonlocal heat content
Volume 270 / 2023
Studia Mathematica 270 (2023), 339-359
MSC: Primary 60J35; Secondary 60G51, 60G52, 35K05.
DOI: 10.4064/sm220831-26-1
Published online: 23 February 2023
Abstract
Let $(p_t)_{t\geq 0}$ be a convolution semigroup of probability measures on $\mathbb R^d$ defined by $$\int _{\mathbb R^d} e^{i\langle \xi ,x\rangle }\, p_t(\mathrm d x)= e^{-t\psi (\xi )}, \quad \ \xi \in \mathbb R^d, $$ and let $\Omega $ be an open subset of $\mathbb R^d$ with finite Lebesgue measure. We consider the quantity $H_{\Omega }(t)= \int _{\Omega }\int _{\Omega -x}p_t( \mathrm d y)\,\mathrm d x$, called the heat content. We study its asymptotic expansion under mild assumptions on $\psi $, in particular in the case of the $\alpha $-stable semigroup.
