A geometric interpretation of the transition density of a symmetric Lévy Process

2011/05/25 by N. Jacob, V. Knopova, Victoria Knopova +8
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1105.5016

arxiv created 2011/05/25 · openalex publication_date 2011/05/25 · arxiv updated 2011/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study for a class of symmetric Lévy processes with state space \rn 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 exp(-δ_t²(x,y)) = \Ff[\frace^-tψp_t(0)](x-y) where \Ff denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: p_t(x)=p_t(0) e^-δ_t2(x,0) where p_t(0) corresponds to a volume term related to √(tψ) and where an "exponential" decay is governed by δ_t². This gives a complete and new geometric, intrinsic interpretation of p_t(x).

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