Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity

Stochastic processes play a key role for mathematically modeling a huge variety of transport problems out of equilibrium. To formulate models of stochastic dynamics the mainstream approach consists in superimposing random fluctuations on a suitable deterministic evolution. These fluctuations are sampled from probability distributions that are prescribed a priori, most commonly as Gaussian or Levy. While these distributions are motivated by (generalised) central limit theorems they are nevertheless unbounded. This property implies the violation of fundamental physical principles such as special relativity and may yield divergencies for basic physical quantities like energy. It is thus clearly never valid in real-world systems by rendering all these stochastic models ontologically unphysical. Here we solve the fundamental problem of unbounded random fluctuations by constructing a comprehensive theoretical framework of stochastic processes possessing finite propagation velocity. Our approach is motivated by the theory of Levy walks, which we embed into an extension of conventional Poisson-Kac processes. Our new theory possesses an intrinsic flexibility that enables the modelling of many different kinds of dynamical features, as we demonstrate by three examples. The corresponding stochastic models capture the whole spectrum of diffusive dynamics from normal to anomalous diffusion, including the striking Brownian yet non Gaussian diffusion, and more sophisticated phenomena such as senescence. Extended Poisson-Kac theory thus not only ensures by construction a mathematical representation of physical reality that is ontologically valid at all time and length scales. It also provides a toolbox of stochastic processes that can be used to model potentially any kind of finite velocity dynamical phenomena observed experimentally.