A note on the generalized heat content for L\'evy processes

Let $\mathbf{X}=\{X_t{\}}_{t\ge 0}$ be a L'{e}vy process in ${\mathbb{R}}^d$ and $\mathrm{\Omega }$ be an open subset of ${\mathbb{R}}^d$ with finite Lebesgue measure. The quantity $H(t)={\int }_{\mathrm{\Omega }}{\mathbb{P}}^x(X_t\in {\mathrm{\Omega }}^c)dx$ is called the heat content. In this article we consider its generalized version $H_g^{\mu }(t)={\int }_{{\mathbb{R}}^d}{\mathbb{E}}^{x}g(X_t)\mu (dx)$, where $g$ is a bounded function and $\mu$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of L'{e}vy processes.