Supersymmetry in random two-velocity processes

Filliger, Roger; Hongler, Max-Olivier

doi:10.1016/j.physa.2003.09.048

Cited by 12 publications

(8 citation statements)

References 27 publications

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“…For applications, the models including 'dichotomic noise', 'random telegraph process', 'binary noise', and so on, as driving terms for deterministic equations, fit in the PDP category. As examples of potential applications, we can quote: reacting-diffusing systems [20], biological dispersal [19], non-Maxwellian equilibriums [1,5,18], filtered telegraph signals [25].…”

Section: Annunziatomentioning

confidence: 99%

On the Action of a Semi-Markov Process on a System of Differential Equations

Annunziato

2012

Mathematical Modelling and Analysis

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We deal with a model equation for stochastic processes that results from the action of a semi-Markov process on a system of ordinary differential equations. The resulting stochastic process is deterministic in pieces, with random changes of the motion at random time epochs. By using classical methods of probability calculus, we first build and discuss the fundamental equation for the statistical analysis, i.e. a Liouville Master Equation for the distribution functions, that is a system of hyperbolic PDE with non-local boundary conditions. Then, as the main contribute to this paper, by using the characteristics' method we recast it to a system of Volterra integral equations with space fluxes, and prove existence and uniqueness of the solution. A numerical experiment for a case of practical application is performed.

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Section: Annunziatomentioning

confidence: 99%

On the Action of a Semi-Markov Process on a System of Differential Equations

Annunziato

2012

Mathematical Modelling and Analysis

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“…We refer to models including the 'dichotomic noise', 'random telegraph process', 'binary noise', and so on, as driving terms for deterministic equations. As examples of such models, we quote: reacting-diffusing systems [17], biological dispersal [16], scattering of radiation [19], non-Maxwellian equilibriums [1,4,6,12,35], filtered telegraph signals [20,32], ecological systems [24], harmonic oscillators [25].…”

Section: Introductionmentioning

confidence: 99%

A Finite Difference Method for Piecewise Deterministic Processes With Memory. Ii

Annunziato

2009

Mathematical Modelling and Analysis

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Abstract. We deal with the numerical scheme for the Liouville Master Equation (LME) of a kind of Piecewise Deterministic Processes (PDP) with memory, analysed in [2]. The LME is a linear system of hyperbolic PDEs, written in non-conservative form, with non-local boundary conditions. The solutions of that equation are time dependent marginal distribution functions whose sum satisfies the total probability conservation law. In [2] the convergence of the numerical scheme, based on the Courant-Isaacson-Rees jointly with a direct quadrature, has been proved under a Courant-Friedrichs-Lewy like (CFL) condition. Here we show that the numerical solution is monotonic under a similar CFL condition. Moreover, we evaluate the conservativity of the total probability for the calculated solution. Finally, an implementation of a parallel algorithm by using the MPI library is described and the results of some performance tests are presented.

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“…is the Chapman-Kolmogorov Eq. for probability densities governing piecewise deterministic evolution models [8]. When time t → ∞ the densities A ± (x.t) will reach a diffusive regime [4], [8] in characterized by a left respectively a right drifted Gaussian:…”

Section: Introductionmentioning

confidence: 99%

“…itself describes the diffusive regime of the probability density P h (y, s) governing a random two-velocities model of the Kac's type [7] with spatially inhomogeneous transitions rates between the velocities. This class of random evolutions is discussed in [8]. For this two-velocities model, the transition probability reads:…”

Section: Introductionmentioning

confidence: 99%

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Quantum Random Walks and Piecewise Deterministic Evolutions

Blanchard

Hongler

2004

Phys. Rev. Lett.

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In the continuous space and time limit, we show that the probability density to find the quantum random walk (QRW) driven by the Hadamard "coin" solves a hyperbolic evolution equation similar to the one obtained for a random two-velocity evolution with spatially inhomogeneous transition rates between the velocity states. In spite of the presence of a nonlinear drift term, it is remarkable that the QRW position can easily be described in simple analytical terms. This allows us to derive the quadratic time dependence of the variance typical for the QRW.

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Supersymmetry in random two-velocity processes

Abstract: We discuss a random two-velocity process on the line with space dependent exogenous drift. For this process, the probability density and the associated "probability current" are shown to be in a supersymmetric relation.

Cited by 12 publications

References 27 publications

On the Action of a Semi-Markov Process on a System of Differential Equations

On the Action of a Semi-Markov Process on a System of Differential Equations

A Finite Difference Method for Piecewise Deterministic Processes With Memory. Ii

Quantum Random Walks and Piecewise Deterministic Evolutions

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