Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part I Basic theory

Abstract

This article introduces the notion of Generalized Poisson-Kac (GPK) processes which generalize the class of "telegrapher's noise dynamics" introduced by Marc Kac in 1974, usingPoissonian 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.

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Cited by in corpus (8)

• Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part III Extensions and applications to kinetic theory and transport
• Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity
• Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part II Irreversibility, Norms and Entropies
• Relativistic analysis of stochastic kinematics
• Series representations for the characteristic function of the multidimensional Markov random flight
• Spectral properties of stochastic processes possessing finite propagation velocity
• In the folds of the Central Limit Theorem: Lévy walks, large deviations and higher-order anomalous diffusion
• Slow diffusion by Markov random flights