A Semigroup Approach to Fractional Poisson Processes

It is well-known that fractional Poisson processes (FPP) constitute an important example of a non-Markovian structure. That is, the FPP has no Markov semigroup associated via the customary Chapman-Kolmogorov equation. This is physically interpreted as the existence of a memory effect. Here, solving a difference-differential equation, we construct a family of contraction semigroups , . If denotes the Banach space of continuous maps from into the Banach space of endomorphisms of a Banach space X, it holds that and is a continuous map from ]0, 1] into . Moreover, becomes the Markov semigroup of a Poisson process.