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

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 $({\delta }_t{)}_{t>0}$. The first family of metrics describes the diagonal term $p_t(0)$; it is induced by the characteristic exponent $\psi$ of the Lévy 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 $$ \exp(-δ_t^2(x,y)) = \Ff[\frac{e^{-tψ}}{p_t(0)}](x-y) $$ where $\Ff$ denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: $p_t(x)=p_t(0)e^{-{\delta }_t^2(x,0)}$ 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 $p_t(x)$.