A geometric interpretation of a symmetric of the transition density Levy process

We study for a class of symmetric Levy processes with state space R^n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t〉o and （δt）t〉o. The first family of metrics describes the diagonal term pt （0）; it is induced by the characteristic exponent ψ of the Levy proces...

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Bibliographic Details
Published in:中国科学：数学英文版 2012, Vol.55 (6), p.1099-1126
Main Author: JACOB Niels KNOPOVA Victorya LANDWEHR Sandra SCHILLING Rene L
Format: Article
Language:English
Subjects:
Levy过程
两个指标
傅里叶变换
几何解释
密度表
对称
特征指数
状态空间
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Summary:We study for a class of symmetric Levy processes with state space R^n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t〉o and （δt）t〉o. The first family of metrics describes the diagonal term pt （0）; it is induced by the characteristic exponent ψ of the Levy process by dr（x, y） = √tψ（x - y）. The second and new family of metrics 6t relates to √tψ through the formula exp（-δ^2t（x,y））=F[e^-tψ/pt（0）]（x-y）,where Y denotes the Fourier transform. Thus we obtain the following ＂Gaussian＂ representation of the tran- sition density： pt（x） = pt（O）e^-δ^2t（x,0） where pt（O） corresponds to a volume term related to √tψ and where an ＂exponential＂ decay is governed by 5t2. This gives a complete and new geometric, intrinsic interpretation of pt（x）.
ISSN:1674-7283
1869-1862