LAUSR

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 $$(T_\alpha )_{\alpha \in ]0,1]}$$(Tα)α∈]0,1], $$T_\alpha =(T_\alpha (t))_{t\ge 0}$$Tα=(Tα(t))t≥0. If $$C([0,\infty [,B(X))$$C([0,∞[,B(X)) denotes the Banach space of continuous maps from $$[0,\infty [$$[0,∞[ into the Banach space of endomorphisms of a Banach space X, it holds that $$T_\alpha \in C([0,\infty [,B(X))$$Tα∈C([0,∞[,B(X)) and $$\alpha \mapsto T_\alpha $$α↦Tα is a continuous map from ]0, 1] into $$C([0,\infty [,B(X))$$C([0,∞[,B(X)). Moreover, $$T_1$$T1 becomes the Markov semigroup of a Poisson process.