A geometric interpretation of the transition density of a symmetric L\'evy Process

We study for a class of symmetric L\'evy 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 $(\delta_t)_{t>0}$. The first family of metrics describes the diagonal term $p_t(0)$; it is induced by the characteristic exponent $\psi$ of the L\'evy process by $d_t(x,y)=\sqrt{t\psi(x-y)}$. The second and new family of metrics $\delta_t$ relates to $\sqrt{t\psi}$ through the formula $$\exp(-\delta_t^2(x,y)) = \Ff[\frac{e^{-t\psi}}{p_t(0)}](x-y)$$ where $\Ff$ denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: $p_t(x)=p_t(0) e^{-\delta_t^2(x,0)}$ where $p_t(0)$ corresponds to a volume term related to $\sqrt{t\psi}$ and where an "exponential" decay is governed by $\delta_t^2$. This gives a complete and new geometric, intrinsic interpretation of $p_t(x)$.