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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.
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참고문헌 (23)
참고문헌 신청
L. Acuna Valverde / 2016 / Heat content estimates over sets of finite perimeter / J. Math. Anal. Appl. 441(1):104~120
L. Acuna Valverde / 2016 / On the one dimensional spectral heat content for stable processes / J. Math. Anal. Appl. 441(1):11~24
L. Acuna Valverde / 2017 / Heat content for stable processes in domains of ${\mathbb{R}}^d$ / J. Geom. Anal. 27(1):492~524
L. Ambrosio / 2000 / Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs / The Clarendon Press;Oxford University Press
D. Applebaum / 2009 / Levy Processes and Stochastic Calculus, second edition, Cambridge Studies in Advanced Mathematics, 116 / Cambridge University Press
L. Acuna Valverde / 2016 / On the one dimensional spectral heat content for stable processes / J. Math. Anal. Appl. 441(1):11~24
L. Acuna Valverde / 2017 / Heat content for stable processes in domains of
L. Ambrosio / 2000 / Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs / The Clarendon Press;Oxford University Press
D. Applebaum / 2009 / Levy Processes and Stochastic Calculus, second edition, Cambridge Studies in Advanced Mathematics, 116 / Cambridge University Press
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