The space-fractional Poisson process

In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^\alpha(t)$, $t\ge 0$, $\alpha \in (0,1]$, are governed by the equations $(\mathrm d/\mathrm dt)p_k^\alpha (t) = -\lambda^\alpha (1-B)^\alpha p_k^\alpha(t)$, where $(1-B)^\alpha$ is the fractional difference operator found in 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.