Ebook: Semi-Markov Random Evolutions
Author: V. Korolyuk A. Swishchuk (auth.)
- Tags: Statistics general, Probability Theory and Stochastic Processes, Integral Equations, Operator Theory, Functional Analysis, Systems Theory Control
- Series: Mathematics and Its Applications 308
- Year: 1995
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-x
Introduction....Pages 1-6
Markov Renewal Processes....Pages 7-26
Phase Merging of Semi-Markov Processes....Pages 27-58
Semi-Markov Random Evolutions....Pages 59-91
Algorithms of Phase Averaging for Semi-Markov Random Evolutions....Pages 93-116
Compactness of Semi-Markov Random Evolutions in the Averaging Scheme....Pages 117-143
Limiting Representations for Semi-Markov Random Evolutions in the Averaging Scheme....Pages 145-173
Compactness of Semi-Markov Random Evolutions in the Diffusion Approximation....Pages 175-206
Stochastic Integral Limiting Representations of Semi-Markov Random Evolutions in the Diffusion Approximation....Pages 207-236
Application of the Limit Theorems to Semi-Markov Random Evolutions in the Averaging Scheme....Pages 237-263
Application of the Diffusion Approximation of Semi-Markov Random Evolutions to Stochastic Systems in Random Media....Pages 265-276
Double Approximation of Random Evolutions....Pages 277-288
Back Matter....Pages 289-310
Content:
Front Matter....Pages i-x
Introduction....Pages 1-6
Markov Renewal Processes....Pages 7-26
Phase Merging of Semi-Markov Processes....Pages 27-58
Semi-Markov Random Evolutions....Pages 59-91
Algorithms of Phase Averaging for Semi-Markov Random Evolutions....Pages 93-116
Compactness of Semi-Markov Random Evolutions in the Averaging Scheme....Pages 117-143
Limiting Representations for Semi-Markov Random Evolutions in the Averaging Scheme....Pages 145-173
Compactness of Semi-Markov Random Evolutions in the Diffusion Approximation....Pages 175-206
Stochastic Integral Limiting Representations of Semi-Markov Random Evolutions in the Diffusion Approximation....Pages 207-236
Application of the Limit Theorems to Semi-Markov Random Evolutions in the Averaging Scheme....Pages 237-263
Application of the Diffusion Approximation of Semi-Markov Random Evolutions to Stochastic Systems in Random Media....Pages 265-276
Double Approximation of Random Evolutions....Pages 277-288
Back Matter....Pages 289-310
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