  We construct admissible circulant Laplacian matrix functions as generators
for strictly increasing random walks on the integer line. These Laplacian
matrix functions refer to a certain class of Bernstein functions. The approach
has connections with biased walks on digraphs. Within this framework, we
introduce a space-time generalization of the Poisson process as a strictly
increasing walk with discrete Mittag-Leffler jumps subordinated to a
(continuous-time) fractional Poisson process. We call this process `{\it
space-time Mittag-Leffler process}'. We derive explicit formulae for the state
probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward)
difference-differential equation of general fractional type. We analyze a
`well-scaled' diffusion limit and obtain a Cauchy problem with a space-time
convolution equation involving Mittag-Leffler densities. We deduce in this
limit the `state density kernel' solving this Cauchy problem. It turns out that
the diffusion limit exhibits connections to Prabhakar general fractional
calculus. We also analyze in this way a generalization of the space-time
fractional Mittag-Leffler process. The approach of construction of good
Laplacian generator functions has a large potential in applications of
space-time generalizations of the Poisson process and in the field of
continuous-time random walks on digraphs.
