Positivity and time behavior of a linear reaction–diffusion system, non‐local in space and time

Khrabustovskyi, Andrii; Stephan, Holger

doi:10.1002/mma.998

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…• A general linear evolution equation that is nonlocal in space and time, including jumps and memory on some domain in R n , can be understood as a limit of a diffusion process (a special Markov process) on a complicated Riemannian manifold (see [9]).…”

Section: Introductionmentioning

confidence: 99%

Memory equations as reduced Markov processes

Stephan¹,

Stephan²

2018

Preprint

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A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we propose an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as a change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realistic modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.

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Section: Introductionmentioning

confidence: 99%

Memory equations as reduced Markov processes

Stephan¹,

Stephan²

2018

Preprint

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“…The papers [6,23] are related to general relativity (according to Wheeler [32], such manifolds can be interpreted as models of the Universe). Some applications of the homogenization theory on manifolds were also presented in [21].…”

Section: Introductionmentioning

confidence: 99%

Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes

Khrabustovskyi

2013

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace-Beltrami operator ∆ ε on the Riemannian manifold M ε (dim M ε = N 2) depending on a small parameter ε > 0. M ε consists of two perforated domains, which are connected by an array of tubes of length q ε . Each perforated domain is obtained by removing from the fixed domain Ω ⊂ R N the system of ε-periodically distributed balls of radius d ε =ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, d ε and q ε . In particular, if the limits lim ε→0 q ε and lim ε→0 (d ε ) N −1 q ε ε −N are positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

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“…[14]) via its Liouville equation. • A general linear evolution equation that is nonlocal in space and time, including jumps and memory on some domain in R n , can be understood as a limit of a diffusion process (a special Markov process) on a complicated Riemannian manifold (see [9]). • The projection of a general Brownian motion (a special Markov process in phase space) on the coordinate space is a diffusion process if the initial velocity is Maxwellian (see [13]).…”

mentioning

confidence: 99%

Memory equations as reduced Markov processes

Stephan¹,

Stephan²

2019

Discrete &Amp; Continuous Dynamical Systems - A

Self Cite

View full text Add to dashboard Cite

A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we propose an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as a change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realistic modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations like the calculation of the equilibrium state. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.

show abstract

Positivity and time behavior of a linear reaction–diffusion system, non‐local in space and time

Cited by 4 publications

References 22 publications

Memory equations as reduced Markov processes

Memory equations as reduced Markov processes

Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes

Memory equations as reduced Markov processes

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