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.

Appendix

Lemma 3.1

Consider the probability measure $\mu (dt)=e^{-t}dt$ on the positive real line. Let be given a sequence $(h_n)_{n\in \mathbb {N}}$ of positive integrable functions such that

(H1):

$h_n (t)\rightarrow h(t)$ almost everywhere as $n\rightarrow \infty$, h integrable, and

(H2):

$\int _0^\infty h_n(t)\mu (dt)\rightarrow \int _0^\infty h(t)\mu (dt)$.

Then $(h_n)_{n\in \mathbb {N}}$ is uniformly integrable and $h_n\rightarrow h$ in $L^1(\mu )$.

Proof

Take $c>0$. Then $h_n1_{\{h_n\le c\}}\rightarrow h1_{\{h\le c\}}$, a.e. by (H1) and $h_n1_{\{h_n\le c\}}\le c$. Thus, Lebesgue’s dominated convergence theorem implies that

$$\begin{aligned} \int _0^\infty h_n(t)1_{\{h_n\le c\}}(t)\mu (dt)\rightarrow \int _0^\infty h(t)1_{\{h\le c\}}(t)\mu (dt). \end{aligned}$$

And (H2) yields

$$\begin{aligned} \int _{\{h_n>c\}}h_n(t)\mu (dt)\rightarrow \int _{\{h>c\}}h(t)\mu (dt). \end{aligned}$$

So that $\limsup _n \int _{\{h_n>c\}}h_n(t)\mu (dt)=\int _{\{h>c\}}h(t)\mu (dt).$ Given $\epsilon >0$, choose $c_0>0$ to have

$$\begin{aligned} \int _{\{h>c_0\}}h(t)\mu (dt)<\epsilon /2 \end{aligned}$$

and there exists $n_0$ such that $\sup _{n\ge n_0}\int _{\{h_n>c_0\}}h_n(t)\mu (dt)<\epsilon .$ Since any finite family of integrable functions is uniformly integrable, so is $(h_1,\ldots ,h_{n_0-1})$, and one obtains the same property for the whole sequence $(h_n)_{n\in \mathbb {N}}$. Finally, the family $(h_n-h)_{n\in \mathbb {N}}$ is uniformly integrable as well, and this sequence converges to 0 a.e. Therefore, $\Vert h_n-h\Vert _1\rightarrow 0$. $\square$