  We analyze some basic issues associated with Generalized Poisson-Kac (GPK)
stochastic processes, starting from the extended notion of the Markovian
condition. The extended Markovian nature of GPK processes is established, and
the implications of this property derived: the associated adjoint formalism for
GPK processes is developed essentially in an analogous way as for the
Fokker-Planck operator associated with Langevin equations driven by Wiener
processes. Subsequently, the regularity of trajectories is addressed: the
occurrence of fractality in the realizations of GPK is a long-term emergent
property, and its implication in thermodynamics is discussed. The concept of
completeness in the stochastic description of GPK is also introduced. Finally,
some observations on the role of correlation properties of noise sources and
their influence on the dynamic properties of transport phenomena are addressed,
using a Wiener model for comparison.
