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

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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