A note on the generalized heat content for L\'{e}vy processes
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Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a L\'{e}vy process in $\mathbb{R}^d$ and $\Omega$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. The quantity $H_{\Omega} (t) = \int_{\Omega}\mathbb{P}^{x} (X_t\in \Omega )\, \mathrm{d} x$ is called the heat content. In this article we consider its generalized version $H_g^\mu (t) = \int_{\mathbb{R}^d}\mathbb{E}^{x} g(X_t)\mu( \ud x )$, 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.
Abstract
Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a L\'{e}vy process in $\mathbb{R}^d$ and $\Omega$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. The quantity $H_{\Omega} (t) = \int_{\Omega}\mathbb{P}^{x} (X_t\in \Omega )\, \mathrm{d} x$ is called the heat content. In this article we consider its generalized version $H_g^\mu (t) = \int_{\mathbb{R}^d}\mathbb{E}^{x} g(X_t)\mu( \ud x )$, 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.
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