Markovian nature, completeness, regularity and correlation properties of Generalized Poisson-Kac processes

TL;DR

This paper explores the fundamental properties of Generalized Poisson-Kac processes, including their Markovian nature, regularity, completeness, and how noise correlations influence their dynamics, with implications for thermodynamics and transport phenomena.

Contribution

It establishes the extended Markovian property of GPK processes and develops an adjoint formalism similar to Fokker-Planck equations, highlighting their unique regularity and correlation features.

Findings

GPK processes are confirmed to be extended Markovian.

Fractality in GPK trajectories emerges over long times.

Correlation properties of noise significantly affect transport dynamics.

Abstract

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.