A note on the generalized heat content for L\'{e}vy processes
Bull. Korean Math. Soc. 2018 Vol. 55, No. 5, 1463-1481
https://doi.org/10.4134/BKMS.b170835
Published online September 1, 2018
Wojciech Cygan, Tomasz Grzywny
Uniwersytet Wroclawski, Politechnika Wroclawska
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 $H_g^\mu (t) = \int_{\mathbb{R}^d}\mathbb{E}^{x} g(X_t)\mu( \mathrm{d} x )$, 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.
Keywords : heat content, isotropic L\'{e}vy process, multivariate regular variation
MSC numbers : 60G51, 60J75, 35K05
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