对称Lévy过程转移密度的几何解释
A geometric interpretation of the transition density of a symmetric Lévy Process
摘要：
本文研究了一类状态空间为$\rn$的对称Lévy过程的转移密度$p_t(x)$关于两个单参数度量族$(d_t)_{t>0}$和$(δ_t)_{t>0}$ .第一类度量描述对角项$p_t(0)$ ;它由$d_t(x,y)=\sqrt{tψ(x-y)}$的Lévy过程的特征指数$ψ$导出。第二和新的度量族$δ_t$通过公式$$ \exp(-δ_t^2(x,y)) = \Ff[\frac{e^{-tψ}}{p_t(0)}](x-y) $$与$\sqrt{tψ}$相关，其中$\Ff$表示傅立叶变换。因此，我们得到了跃迁密度的以下“高斯”表示：$p_t(x)=p_t(0) e^{-δ_t^2(x,0)}$其中$p_t(0)$对应于与$\sqrt{tψ}$相关的体积项，并且其中“指数”衰减由$δ_t^2$控制。这给出了$p_t(x)$的一个完整的和新的几何的、内在的解释。
英文摘要：
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 $d_t(x,y)=\sqrt{tψ(x-y)}$. The second and new family of metrics $δ_t$ relates to $\sqrt{tψ}$ through the formula $$ \exp(-δ_t^2(x,y)) = \Ff[\frac{e^{-tψ}}{p_t(0)}](x-y) $$ where $\Ff$ denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: $p_t(x)=p_t(0) e^{-δ_t^2(x,0)}$ where $p_t(0)$ corresponds to a volume term related to $\sqrt{tψ}$ and where an "exponential" decay is governed by $δ_t^2$. This gives a complete and new geometric, intrinsic interpretation of $p_t(x)$.
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