On stochastic differential equations with random delay

Krapivsky, P. L.; Luck, J. M.; Mallick, Kirone

doi:10.1088/1742-5468/2011/10/p10008

Cited by 10 publications

(6 citation statements)

References 32 publications

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“…( 5) for the micropolar lattice at low frequencies is quadratic with respect to the wavenumber i.e., Ω ≈ 3 √ K (2) Q 2 in contrast to the 1D harmonic lattice where Ω ≈ Q. Furthermore, for the micropolar lattice, the quasi-extended modes at higher frequencies may influence the energy spreading, as it was shown for example in [12,52] where additional extended modes were found either due to symmetries or resonances. As far as point (ii) is concerned, the results of Figs.…”

Section: Dynamics Of the Systemmentioning

confidence: 97%

Wave propagation in a strongly disordered 1D phononic lattice supporting rotational waves

Ngapasare,

Theocharis,

Richoux

et al. 2020

Preprint

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We investigate the dynamical properties of a strongly disordered micropolar lattice made up of cubic block units. This phononic lattice model supports both transverse and rotational degrees of freedom hence its disordered variant posses an interesting problem as it can be used to model physically important systems like beam-like microstructures. Different kinds of single site excitations (momentum or displacement) on the two degrees of freedom are found to lead to different energy transport both superdiffusive and subdiffusive. We show that the energy spreading is facilitated both by the low frequency extended waves and a set of high frequency modes located at the edge of the upper branch of the periodic case for any initial condition. However, the second moment of the energy distribution strongly depends on the initial condition and it is slower than the underlying one dimensional harmonic lattice (with one degree of freedom). Finally, a limiting case of the micropolar lattice is studied where Anderson localization is found to persist and no energy spreading takes place.

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Section: Dynamics Of the Systemmentioning

confidence: 97%

Wave propagation in a strongly disordered 1D phononic lattice supporting rotational waves

Ngapasare,

Theocharis,

Richoux

et al. 2020

Preprint

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“…On the other hand, in fuzzy delay differential equations, uncertainty is driven by fuzzy processes; see [29] for instance. In any of these approaches, the delay might even be considered random; see [30,31].…”

Section: Introductionmentioning

confidence: 99%

Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

López

Jornet

2020

Mathematics

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This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay τ > 0 , by adding a random forcing term f ( t ) that varies with time: x ′ ( t ) = a x ( t ) + b x ( t − τ ) + f ( t ) , t ≥ 0 , with initial condition x ( t ) = g ( t ) , − τ ≤ t ≤ 0 . The coefficients a and b are assumed to be random variables, while the forcing term f ( t ) and the initial condition g ( t ) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L p -solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay τ tends to 0, the random delay equation tends in L p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.

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“…Finally, it must be pointed out that randomness is directly introduced in the delay instead of coefficients and/or forcing term in order to account for uncertainties associated to the time instant in which relevant factors determining the output of the mathematical model under study take place. Examples in this regard can be found in [29][30][31], for example.…”

Section: Introductionmentioning

confidence: 99%

$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

2019

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In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay τ > 0: x (t) = ax(t) + bx(t − τ), t ≥ 0, with initial condition x(t) = g(t), −τ ≤ t ≤ 0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. By using L p-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an L p-solution too. An analysis of L p-convergence when the delay τ tends to 0 is also performed in detail.

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On stochastic differential equations with random delay

Cited by 10 publications

References 32 publications

Wave propagation in a strongly disordered 1D phononic lattice supporting rotational waves

Wave propagation in a strongly disordered 1D phononic lattice supporting rotational waves

Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

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