  In this paper we consider a scalar transport equation with constant
coefficients on domains with discrete space and continuous, discrete or general
time. We show that on all these underlying domains solutions of the transport
equation can conserve sign and integrals both in time and space. Detailed
analysis reveals that, under some initial conditions, the solutions correspond
to counting stochastic processes and related probability distributions.
Consequently, the transport equation could generate various modifications of
these processes and distributions and provide some insights into corresponding
convergence questions. Possible applications are suggested and discussed.
