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We classify L\'evy processes according to the solution spaces of theassociated parabolic PIDEs. This classification reveals structuralcharacteristics of the processes and is relevant for applications such as forsolving PIDEs numerically for pricing options in L\'evy models.  The classification is done via the Fourier transform i.e. via the symbol ofthe process. We define the Sobolev index of a L\'evy process by a certaingrowth condition on the symbol. It follows that for L\'evy processes withSobolev index $\alpha$ the corresponding evolution problem has a unique weaksolution in the Sobolev-Slobodeckii space $H^{\alpha/2}$. We show that thisclassification applies to a wide range of processes. Examples are the Brownianmotion 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 shedslight on the structural implication of the classification. More precisely, wediscuss the Sobolev index as an indicator of the smoothness of the distributionand of the variation of the paths of the process. This highlights the relationbetween the $p$-variation of the paths and the degree of smoothing effect thatstems from the distribution.

Sobolev index: A classification of L\'evy processes