Some Considerations on the Structure of Transition Densities of Symmetric Lévy Processes

For a class of symmetric Lévy processes (Yt)t≥0 with characteristic exponent ψ we show that νt = e − 1 t /p 1 t (0), t > 0, gives rise to an additive process (Xt)t≥0 with t-dependent characteristic exponent − ∂ ∂t ln ( p 1 t (ξ)/p 1 t (0) ) where (pt)t>0 are the transition densities of (Yt)t≥0. We estimate (from above and below) pt in terms of two metrics dψ,t and δψ,t, dψ,t controlling pt(0) and δψ,t the spatial decay, and we prove that the transition density πt,0 of PXt−X0 is controlled by δψ, 1 t and d ψ, 1 t now with δ ψ, 1 t controlling πt,0(0) and dψ, 1 t the spatial decay.