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 (Tα)α∈]0,1], Tα=(Tα(t))t≥0. If C([ 0 , ∞[ , B(X)) denotes the Banach space of continuous maps from [ 0 , ∞[ into the Banach space of endomorphisms of a Banach space X, it holds that Tα∈ C([ 0 , ∞[ , B(X)) and α↦ Tα is a continuous map from ]0, 1] into C([ 0 , ∞[ , B(X)). Moreover, T1 becomes the Markov semigroup of a Poisson process.