A Semigroup Approach to Fractional Poisson Processes

Abstract

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 \((T_\alpha )_{\alpha \in ]0,1]}\), \(T_\alpha =(T_\alpha (t))_{t\ge 0}\). If \(C([0,\infty [,B(X))\) denotes the Banach space of continuous maps from \([0,\infty [\) into the Banach space of endomorphisms of a Banach space X, it holds that \(T_\alpha \in C([0,\infty [,B(X))\) and \(\alpha \mapsto T_\alpha \) is a continuous map from ]0, 1] into \(C([0,\infty [,B(X))\). Moreover, \(T_1\) becomes the Markov semigroup of a Poisson process.