A geometric interpretation of the transition density of a symmetric Lévy process |
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作者姓名: | JACOB Niels KNOPOVA Victorya LANDWEHR Sandra SCHILLING René L. |
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作者单位: | Department of Mathematics, Swansea University, Singleton Park, Swansea SA28PP, UK;V. M. Glushkov Institute of Cybernetics NAS of Ukraine, 03187, Kiev, Ukraine;Heinrich Heine University Düsseldorf, German Diabetes Center at the Heinrich Heine University Düsseldorf, Leibniz Center for Diabetes Research, Institute of Biometrics and Epidemiology, Auf’m Hennekamp 65, 40225 Düsseldorf, Germany;Institut für Mathematische Stochastik, Technische Universitt Dresden, 01062 Dresden, Germany |
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摘 要: | We study for a class of symmetric Lévy processes with state space R n the transition density pt(x) in terms of two one-parameter families of metrics, (dt)t>0 and (δt)t>0. The first family of metrics describes the diagonal term pt(0); it is induced by the characteristic exponent ψ of the Lévy process by dt(x, y) = 1/2tψ(x-y). The second and new family of metrics δt relates to 1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt(x) = pt(0)e- δ2t (x,0) where pt(0) corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt(x).
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关 键 词: | transition function estimates Lévy processes metric measure spaces heat kernel bounds infinitely divisible distributions self-reciprocal distributions |
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