Stochastic foundations of undulatory transport phenomena. Generalized Poisson-Kac processes. Part I basic theory

This article introduces the notion of generalized Poisson–Kac (GPK) processes which generalize the class of ‘telegrapher’s noise dynamics’ introduced by Kac (1974 Rocky Mount. J. Math. 4 497) 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/long-distance 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