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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). 相似文献
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