On the parabolic generator of a general one-dimensional Levy process

The purpose of this note is twofold. Firstly to complete a recent accumulation of results concerning extended version of Itô's formula for any one dimensional Lévy processes, X. Secondly, we use the latter to characterise the parabolic generator of X A := {(f, g) : f(X., ·) - ∫0 g(Xs, s)ds is a local martingale}. We also establish a necessary condition for a pair of functions to be in the domain of the parabolic generator when X has a Gaussian component.