This article introduces the notion of Generalized Poisson-Kac (GPK) processes which generalize the class of "telegrapher's noise dynamics" introduced by Marc Kac [39] in 1974, using Poissonian stochastic perturbations. In GPK processes the stochastic perturbation acts as a switching amongst a set of stochastic velocity vectors controlled by a Markov-chain dynamics. GPK processes possess trajectory regularity (almost everywhere) and asymptotic Kac limit, namely the convergence towards Brownian motion (and to stochastic dynamics driven by Wiener perturbations), which characterizes also the long-term/longdistance properties of these processes. In this article we introduce the structural properties of GPK processes, leaving all the physical implications to part II and part III [80,81].In statistical physics of gases, liquids, polymers and colloids (soft-matter physics, for short), the use of Brownian motion and Wiener processes can be regarded as the legacy of a large-number ansatz, wherein the influence of a manifold of small contributions, uniquely characterized by their mean and variance justifies the application of stochastic perturbations characterized by uncorrelated increments distributed in a normal way, which is precisely the definition of a Wiener process [6,7,8].The statistical characterization of microdynamic equations written in the form of Wiener-Langevin stochastic dynamics leads (using any representation of the stochastic integrals in the meaning of Ito, Stratonovich, Klimontovich, etc.) to parabolic (forward Fokker-Planck) equations for the probability density function, in which a deterministic drift v(x), and a tensorial diffusivity D can be always identified, possessing the structure of a second-order advection-diffusion equation.The analogy between Fokker-Planck equations and advection-diffusion problems permits to identify the overall probability flux J p (x, t) associated with the probability density p(x, t) as
Basic principlesThe structure of one-dimensional Poisson-Kac processes, reviewed in the Appendix, contains the germs for its generalization in higher dimensions. Three basic principles can be enucleated out of it, that can guide the development of a simple and consistent stochastic theory of undulatory transport phenomena. These principles are:• The principle of stochastic reality;• The principle of the primitive variables;• The principle of the asymptotic Kac convergence.Below, their meaning and importance is outlined and discussed. 7
