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 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.
Highlights
The investigation of micro- and nanoscale physics [1] as well as the extension of statistical physical concepts to new phenomelogies involving active matter and living beings [2,3] have stimulated the development of more refined descriptions of stochastic processes, accounting for background and thermal or quantum fluctuations, aimed at deriving their statistical features and long-term, in some cases anomalous [4], scaling properties [5,6]
We thoroughly investigate the fluctuation spectra of Generalized Poisson-Kac processes (GPK) [15,16,17,18] and Lévy Walks (LW) [19,20,21,22], as these processes represent two of the main classes of stochastic models characterized by bounded propagation velocity
Owing to the analysis developed by Fedotov [29], subsequently elaborated in [30,31,32,33] and extended in a unitary theory of stochastic processes possessing bounded velocity in [34], a complete statistical description of an LW involves a system of partial probability densities, as for GPK processes
Summary
The investigation of micro- and nanoscale physics [1] as well as the extension of statistical physical concepts to new phenomelogies involving active matter and living beings [2,3] have stimulated the development of more refined descriptions of stochastic processes, accounting for background and thermal or quantum fluctuations, aimed at deriving their statistical features and long-term, in some cases anomalous [4], scaling properties [5,6]. This represents an extension of the original model proposed by Kac [15] of a stochastic process driven by the parity of a Poisson counting process, which has been the subject of intense investigation in the past as a prototype of a non-Markovian stochastic motion driven by bounded, dichotomous, and colored noise [24,25,26,27,28] We choose these two classes of processes because they are subjected to a common and complete statistical description in terms of the partial probability densities parametrized with respect to the internal variables of the process (for GPK, the velocity direction; for LWs, the transition age). This provides an interesting and exactly solvable example of a prototypical sub-quantum theory, in line with David Bohm [38,39], whose theory’s emergent properties coincide with the classical Schrödinger equation
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