A geometric interpretation of the transition density of a symmetric Lévy process
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Contributors
Abstract
We study for a class of symmetric Lévy processes with state space ℝ n the transition density p t (x) in terms of two one-parameter families of metrics, (d t) t>0 and (δ t) t>0. The first family of metrics describes the diagonal term p t (0); it is induced by the characteristic exponent ψ of the Lévy process by d t(x,y)=√tψ(x-y). The second and new family of metrics δ t relates to √tψ through the formula,where F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density:corresponds to a volume term related to p t(0) and where an √tψ. This gives a complete and new geometric, intrinsic interpretation of p t(x).
Details
Original language | English |
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Pages (from-to) | 1099-1126 |
Number of pages | 28 |
Journal | Science China : Mathematics |
Volume | 55 |
Issue number | 6 |
Publication status | Published - Jun 2012 |
Peer-reviewed | Yes |
Keywords
ASJC Scopus subject areas
Keywords
- heat kernel bounds, infinitely divisible distributions, Lévy processes, metric measure spaces, self-reciprocal distributions, transition function estimates