Asymptotic expansion of the nonlocal heat content

Let $\mathbf{X}=\{X_t{\}}_{t\ge 0}$ be a L'evy process in ${\mathbb{R}}^d$ and $\mathrm{\Omega }$ be an open subset of ${\mathbb{R}}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=\int_{\Omega} $\mathbb{P}$^x (X_t\in\Omega^c) , $\mathrm{d}$x$ which is called the heat content. We study its asymptotic expansion for isotropic $\alpha$-stable L'evy processes and more general L'evy processes, under mild assumptions on the characteristic exponent.