Generalized Poisson Kac Processes Research Articles

Spectral Properties of Stochastic Processes Possessing Finite Propagation Velocity.

This article investigates the spectral structure of the evolution operators associated with the statistical description of stochastic processes possessing finite propagation velocity. Generalized Poisson–Kac processes and Lévy walks are explicitly considered as paradigmatic examples of regular and anomalous dynamics. A generic spectral feature of these processes is the lower boundedness of the real part of the eigenvalue spectrum that corresponds to an upper limit of the spectral dispersion curve, physically expressing the relaxation rate of a disturbance as a function of the wave vector. We also analyze Generalized Poisson–Kac processes possessing a continuum of stochastic states parametrized with respect to the velocity. In this case, there is a critical value for the wave vector, above which the point spectrum ceases to exist, and the relaxation dynamics becomes controlled by the essential part of the spectrum. This model can be extended to the quantum case, and in fact, it represents a simple and clear example of a sub-quantum dynamics with hidden variables.

Covariance and Spinorial Statistical Description of Simple Relativistic Stochastic Kinematics

It is shown that Generalized Poisson–Kac processes are closed with respect to Lorentz transformations, providing a class of covariant kinematic processes. The transformation properties of the associated partial probability densities (waves) display spinorial character in a probability space, and their spinorial character is intrinsically related to the parametrization of the internal degrees of freedom of the process. Parity function analysis associated with the bias induced in the partial-wave recombination process by a Lorentz boost, indicates a symmetry breaking in the recombination dynamics. In an inertial reference frame moving with constant velocity [Formula: see text] with respect to the rest frame of the process, stochastic fluctuations are progressively damped out till complete suppression in the limit for [Formula: see text].

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 (Giona et al 2016a J. Phys. A: Math. Theor., 2016b J. Phys. A: Math. Theor.).

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part II Irreversibility, norms and entropies

In this second part, we analyze the dissipation properties of generalized Poisson–Kac (GPK) processes, considering the decay of suitable L2-norms and the definition of entropy functions. In both cases, consistent energy dissipation and entropy functions depend on the whole system of primitive statistical variables, the partial probability density functions , while the corresponding energy dissipation and entropy functions based on the overall probability density do not satisfy monotonicity requirements as a function of time. These results provide new insights on the theory of Markov operators associated with irreversible stochastic dynamics. Examples from chaotic advection (standard map coupled to stochastic GPK processes) illustrate this phenomenon. Some complementary physical issues are also addressed: the ergodicity breaking in the presence of attractive potentials, and the use of GPK perturbations to mollify stochastic field equations.

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part III extensions and applications to kinetic theory and transport

This third part extends the theory of Generalized Poisson–Kac (GPK) processes to nonlinear stochastic models and to a continuum of states. Nonlinearity is treated in two ways: (i) as a dependence of the parameters (intensity of the stochastic velocity, transition rates) of the stochastic perturbation on the state variable, similarly to the case of nonlinear Langevin equations, and (ii) as the dependence of the stochastic microdynamic equations of motion on the statistical description of the process itself (nonlinear Fokker–Planck–Kac models). Several numerical and physical examples illustrate the theory. Gathering nonlinearity and a continuum of states, GPK theory provides a stochastic derivation of the nonlinear Boltzmann equation, furnishing a positive answer to the Kac’s program in kinetic theory. The transition from stochastic microdynamics to transport theory within the framework of the GPK paradigm is also addressed.

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

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.

Variational principles and Lagrangian functions for stochastic processes and their dissipative statistical descriptions

A variational Lagrangian formulation for stochastic processes and for the evolution equations of the associated probability density functions is developed. Particular attention is dedicated to Poisson–Kac processes possessing finite propagation velocity. The variational formulation in terms of Lagrangian and Hamiltonian densities permits to address different forms of “reversibility” characterizing these processes with respect to processes driven by nowhere differentiable Wiener fluctuations, and associated with the concepts of dynamic and statistical reversibility. The latter property, i.e. the statistical reversibility, implies that the extended Markov operator associated with Poisson–Kac and Generalized Poisson–Kac processes forms a group, continuously parametrized with respect to time.

Generalized Poisson–Kac Processes: Basic Properties and Implications in Extended Thermodynamics and Transport

Abstract We introduce a new class of stochastic processes in R n , $${{\rm{\mathbb R}}^n}, $$ referred to as generalized Poisson–Kac (GPK) processes, that generalizes the Poisson–Kac telegrapher’s random motion in higher dimensions. These stochastic processes possess finite propagation velocity, almost everywhere smooth trajectories, and converge in the Kac limit to Brownian motion. GPK processes are defined by coupling the selection of a bounded velocity vector from a family of N distinct ones with a Markovian dynamics controlling probabilistically this selection. This model can be used as a probabilistic tool for a stochastically consistent formulation of extended thermodynamic theories far from equilibrium.