Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2018; 55(5): 1463-1481

Online first article March 15, 2018      Printed September 1, 2018

https://doi.org/10.4134/BKMS.b170835

Copyright © The Korean Mathematical Society.

A note on the generalized heat content for L\'{e}vy processes

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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