On the rôle of singular functions in extending the probabilistic symbol to its most general class

The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information concerning the stochastic process. For a Lévy process, one obtains the characteristic exponent while the symbol of a (rich) Feller process coincides with the classical symbol which is well known from the theory of pseudodifferential operators. Leaving these classes behind, the most general class of processes for which the symbol still exists are Lévy-type processes. It has been an open question, whether further generalizations are possible within the framework of Markov processes. We answer this question in the present article: within the class of Hunt semimartingales, Lévy-type processes are exactly those for which the probabilistic symbol exists. Leaving Hunt behind, one can construct processes admitting a symbol. However, we show, that the applicability of the symbol might be lost for these processes. Surprisingly, in our proofs the upper and lower Dini derivatives corresponding to certain singular functions play an important rôle.