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Keywords = Poisson–Kac processes

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25 pages, 988 KB  
Article
Spectral Properties of Stochastic Processes Possessing Finite Propagation Velocity
by Massimiliano Giona, Andrea Cairoli, Davide Cocco and Rainer Klages
Entropy 2022, 24(2), 201; https://doi.org/10.3390/e24020201 - 28 Jan 2022
Cited by 2 | Viewed by 3292
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Universality in Anomalous Transport Processes)
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19 pages, 1395 KB  
Article
Swelling and Drug Release in Polymers through the Theory of Poisson–Kac Stochastic Processes
by Alessandra Adrover, Claudia Venditti and Massimiliano Giona
Gels 2021, 7(1), 32; https://doi.org/10.3390/gels7010032 - 22 Mar 2021
Cited by 9 | Viewed by 2657
Abstract
Experiments on swelling and solute transport in polymeric systems clearly indicate that the classical parabolic models fail to predict typical non-Fickian features of sorption kinetics. The formulation of moving-boundary transport models for solvent penetration and drug release in swelling polymeric systems is addressed [...] Read more.
Experiments on swelling and solute transport in polymeric systems clearly indicate that the classical parabolic models fail to predict typical non-Fickian features of sorption kinetics. The formulation of moving-boundary transport models for solvent penetration and drug release in swelling polymeric systems is addressed hereby employing the theory of Poisson–Kac stochastic processes possessing finite propagation velocity. The hyperbolic continuous equations deriving from Poisson–Kac processes are extended to include the description of the temporal evolution of both the Glass–Gel and the Gel–Solvent interfaces. The influence of polymer relaxation time on sorption curves and drug release kinetics is addressed in detail. Full article
(This article belongs to the Special Issue Gels: 6th Anniversary)
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17 pages, 918 KB  
Article
Space-Time Inversion of Stochastic Dynamics
by Massimiliano Giona, Antonio Brasiello and Alessandra Adrover
Symmetry 2020, 12(5), 839; https://doi.org/10.3390/sym12050839 - 20 May 2020
Viewed by 2466
Abstract
This article introduces the concept of space-time inversion of stochastic Langevin equations as a way of transforming the parametrization of the dynamics from time to a monotonically varying spatial coordinate. A typical physical problem in which this approach can be fruitfully used is [...] Read more.
This article introduces the concept of space-time inversion of stochastic Langevin equations as a way of transforming the parametrization of the dynamics from time to a monotonically varying spatial coordinate. A typical physical problem in which this approach can be fruitfully used is the analysis of solute dispersion in long straight tubes (Taylor-Aris dispersion), where the time-parametrization of the dynamics is recast in that of the axial coordinate. This allows the connection between the analysis of the forward (in time) evolution of the process and that of its exit-time statistics. The derivation of the Fokker-Planck equation for the inverted dynamics requires attention: it can be deduced using a mollified approach of the Wiener perturbations “a-la Wong-Zakai” by considering a sequence of almost everywhere smooth stochastic processes (in the present case, Poisson-Kac processes), converging to the Wiener processes in some limit (the Kac limit). The mathematical interpretation of the resulting Fokker-Planck equation can be obtained by introducing a new way of considering the stochastic integrals over the increments of a Wiener process, referred to as stochastic Stjelties integrals of mixed order. Several examples ranging from stochastic thermodynamics and fractal-time models are also analyzed. Full article
(This article belongs to the Special Issue Advances in Stochastic Differential Equations)
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