Semi-Markov Random Evolutions

Korolyuk, V. S.; Swishchuk, Anatoliy

doi:10.1007/978-94-011-1010-5

Cited by 95 publications

(34 citation statements)

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“…We consider a semi-Markov process x ( t ), t ≥ 0, that is regular and uniformly ergodic [8] with stationary distribution π ( B ), B ∈ ᑲ, that satisfies the relation [9] …”

Section: Statement Of the Problemmentioning

confidence: 99%

Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation

Chabanyuk

2007

Ukr Math J

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“…We consider a semi-Markov process x ( t ), t ≥ 0, that is regular and uniformly ergodic [8] with stationary distribution π ( B ), B ∈ ᑲ, that satisfies the relation [9] …”

Section: Statement Of the Problemmentioning

confidence: 99%

Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation

Chabanyuk

2007

Ukr Math J

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“…(MEF) a Markov Evolution Family or a Markov Multiplicative Cocycle [6,7] given as a two-parameter family of Cauchy matrices {X ε (t, s), t ≥ s ≥ 0} satisfying a linear differential equation in R n :…”

Section: The Model Assumptions and Notationmentioning

confidence: 99%

On stochastic stability of Markov evolution associated with impulse Markov dynamical systems

Korolyuk

Carkovs

2008

Theor. Probability and Math. Statist.

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Abstract. This paper deals with the family of Cauchy matrices of a linear differential equation dependent on a step Markov process and an impulse type dynamical system rapidly switched by the above process. Applying the stochastic and deterministic averaging procedures according to the invariant measures of the Markov process one achieves a simpler linear differential equation dependent on simpler dynamical systems such as an ordinary differential equation, a differential equation with the right hand side switched by a merger Markov process or a stochastic Itô differential equation. It is proved that under some hypotheses one may successfully apply these resulting evolution families not only to analyzing the initial family on an arbitrary finite time interval but also to describing a time asymptotic of this family.

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“…x a b ∈ Equation (3) determines the random evolution of the particle in the alternating Markov medium { ( ) ξt ; t ≥ 0} [6,7]. So, x(t) is the well-known one-dimensional telegraph process [1,2].…”

Section: Mathematical Modelingmentioning

confidence: 99%

Stationary Distribution of Random Motion with Delay in Reflecting Boundaries

Pogorui¹,

Rodríguez‐Dagnino

2010

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In this paper we study a continuous time random walk in the line with two boundaries [a,b], a < b. The particle can move in any of two directions with different velocities v 1 and v 2 . We consider a special type of boundary which can trap the particle for a random time. We found closed-form expressions for the stationary distribution of the position of the particle not only for the alternating Markov process but also for a broad class of semi-Markov processes.

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Semi-Markov Random Evolutions

Cited by 95 publications

References 0 publications

Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation

Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation

On stochastic stability of Markov evolution associated with impulse Markov dynamical systems

Stationary Distribution of Random Motion with Delay in Reflecting Boundaries

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