  We classify L\'evy processes according to the solution spaces of the
associated parabolic PIDEs. This classification reveals structural
characteristics of the processes and is relevant for applications such as for
solving PIDEs numerically for pricing options in L\'evy models.
  The classification is done via the Fourier transform i.e. via the symbol of
the process. We define the Sobolev index of a L\'evy process by a certain
growth condition on the symbol. It follows that for L\'evy processes with
Sobolev index $\alpha$ the corresponding evolution problem has a unique weak
solution in the Sobolev-Slobodeckii space $H^{\alpha/2}$. We show that this
classification applies to a wide range of processes. Examples are the Brownian
motion with or without drift, generalised hyperbolic (GH), CGMY and (semi)
stable L\'evy processes.
  A comparison of the Sobolev index with the Blumenthal-Getoor index sheds
light on the structural implication of the classification. More precisely, we
discuss the Sobolev index as an indicator of the smoothness of the distribution
and of the variation of the paths of the process. This highlights the relation
between the $p$-variation of the paths and the degree of smoothing effect that
stems from the distribution.
