Spectral properties of stochastic processes possessing finite propagation velocity

TL;DR

This paper explores the spectral characteristics of stochastic processes with finite propagation speed, focusing on Poisson-Kac processes and Lévy walks, revealing bounds on relaxation rates and spectral transitions relevant to classical and quantum dynamics.

Contribution

It provides a detailed spectral analysis of processes with finite propagation velocity, including new insights into their eigenvalue spectra and implications for quantum analogs.

Findings

Eigenvalue spectrum is lower-bounded in these processes.

Existence of a critical wavevector where the spectrum transitions from point to essential.

Spectral properties inform the relaxation dynamics of stochastic processes.

Abstract

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 L\'evy 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, corresponding to an upper limit for the spectral dispersion curve, physically expressing the relaxation rate of a disturbance as a function of the wave vector. We analyze also Generalized Poisson-Kac processes possessing a continuum of stochastic states parametrized with respect to the velocity. In this case, there exists a critical value of the wavevector 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 point of fact, it represents a simple and highlighting example of a sub-quantum dynamics with hidden variables.