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Semi-Markov Random Evolutions

  • Book
  • © 1995

Overview

Authors:
  1. V. Korolyuk

    1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

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  2. A. Swishchuk

    1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Mathematics and Its Applications (MAIA, volume 308)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction

    • V. Korolyuk, A. Swishchuk
    Pages 1-6

  3. Markov Renewal Processes

    • V. Korolyuk, A. Swishchuk
    Pages 7-26

  4. Phase Merging of Semi-Markov Processes

    • V. Korolyuk, A. Swishchuk
    Pages 27-58

  5. Semi-Markov Random Evolutions

    • V. Korolyuk, A. Swishchuk
    Pages 59-91

  6. Algorithms of Phase Averaging for Semi-Markov Random Evolutions

    • V. Korolyuk, A. Swishchuk
    Pages 93-116

  7. Compactness of Semi-Markov Random Evolutions in the Averaging Scheme

    • V. Korolyuk, A. Swishchuk
    Pages 117-143

  8. Limiting Representations for Semi-Markov Random Evolutions in the Averaging Scheme

    • V. Korolyuk, A. Swishchuk
    Pages 145-173

  9. Compactness of Semi-Markov Random Evolutions in the Diffusion Approximation

    • V. Korolyuk, A. Swishchuk
    Pages 175-206

  10. Stochastic Integral Limiting Representations of Semi-Markov Random Evolutions in the Diffusion Approximation

    • V. Korolyuk, A. Swishchuk
    Pages 207-236

  11. Application of the Limit Theorems to Semi-Markov Random Evolutions in the Averaging Scheme

    • V. Korolyuk, A. Swishchuk
    Pages 237-263

  12. Application of the Diffusion Approximation of Semi-Markov Random Evolutions to Stochastic Systems in Random Media

    • V. Korolyuk, A. Swishchuk
    Pages 265-276

  13. Double Approximation of Random Evolutions

    • V. Korolyuk, A. Swishchuk
    Pages 277-288

  14. Back Matter

    Pages 289-310
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Keywords

About this book

The evolution of systems in random media is a broad and fruitful field for the applica­ tions of different mathematical methods and theories. This evolution can be character­ ized by a semigroup property. In the abstract form, this property is given by a semigroup of operators in a normed vector (Banach) space. In the practically boundless variety of mathematical models of the evolutionary systems, we have chosen the semi-Markov ran­ dom evolutions as an object of our consideration. The definition of the evolutions of this type is based on rather simple initial assumptions. The random medium is described by the Markov renewal processes or by the semi­ Markov processes. The local characteristics of the system depend on the state of the ran­ dom medium. At the same time, the evolution of the system does not affect the medium. Hence, the semi-Markov random evolutions are described by two processes, namely, by the switching Markov renewal process, which describes the changes of the state of the external random medium, and by the switched process, i.e., by the semigroup of oper­ ators describing the evolution of the system in the semi-Markov random medium.

Authors and Affiliations

  • Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

    V. Korolyuk, A. Swishchuk

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