The Space-Fractional Poisson Process

In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^{\alpha }(t)$, $t>0$, $\alpha \in (0,1]$, are governed by the equations $(\mathrm{d}\mathrm{/}\mathrm{d}t)p_k(t)=-{\lambda }^{\alpha }(1-B)p_k^{\alpha }(t)$, where $(1-B{)}^{\alpha }$ is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions $p_k^{\alpha }(t)$, the probability generating functions $G_{\alpha }(u,t)$, which are also expressed as distributions of the minimum of i.i.d.\ uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space-time fractional Poisson process of which we give the explicit distribution.