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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book illustrates a number of asymptotic and analytic approaches applied for the study of random evolutionary systems, and considers typical problems for specific examples. In this case, constructive mathematical models of natural processes are used, which more realistically describe the trajectories of diffusion-type processes, rather than those of the Wiener process. We examine models where particles have some free distance between two consecutive collisions. At the same time, we investigate two cases: the Markov evolutionary system, where the time during which the particle moves towards some direction is distributed exponentially with intensity parameter lambda; and the semi-Markov evolutionary system, with arbitrary distribution of the switching process. Thus, the models investigated here describe the motion of particles with a finite speed and the proposed random evolutionary process with characteristics of a natural physical process: free run and finite propagation speed. In the proposed models, the number of possible directions of evolution can be finite or infinite.
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Veröffentlichungsjahr: 2023
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Contents
Cover
Dedication Page
Title Page
Copyright Page
Preface
Introduction
Chapter 1. Multidimensional Models of Kac Type
1.1. Definitions and basic properties
1.2. Moments of evolutionary process
1.3. Systems of Kolmogorov equations
1.4. Evolutionary operator and theorem about weak convergence to the measure of the Wiener process
Chapter 2. Symmetry of Markov Random Evolutionary Processes in
R
n
2.1. Symmetrization: definition and properties
2.2. Examples of symmetric distributions in
R
n
and distributions on
n
+ 1-hedra
Chapter 3. Hyperparabolic Equations, Integral Equation and Distribution for Markov Random Evolutionary Processes
3.1. Hyperparabolic equations and methods of solving Cauchy problems
3.2. Analytical solution of a hyperparabolic equation with real-analytic initial conditions
3.3. Integral representation of the hyperparabolic equation
3.4. Distribution function of evolutionary process
Chapter 4. Fading Markov Random Evolutionary Process
4.1. Definition of fading Markov random evolutionary process, its moments and limit distribution
4.2. Integral equation for a function from the fading random evolutionary process
4.3. Equations in partial derivatives for a function of the fading random evolutionary process
Chapter 5. Two Models of the Evolutionary Process
5.1. Evolution on a complex plane
5.2. Evolution with infinitely many directions
Chapter 6. Diffusion Process with Evolution and Its Parameter Estimation
6.1. Asymptotic diffusion environment
6.2. Approximation of a discrete Markov process in asymptotic diffusion environment
6.3. Parameter estimation of the limit process
Chapter 7. Filtration of Stationary Gaussian Statistical Experiments
7.1. Introduction
7.2. Stochastic difference equation of the process of filtration
7.3. Coefficient of filtration
7.4. Equation of optimal filtration
7.5. Characterization of a filtered signal
Chapter 8. Adapted Statistical Experiments with Random Change of Time
8.1. Introduction
8.2. Statistical experiments and evolutionary processes
8.3. Stochastic dynamics of statistical experiments
8.4. Adapted statistical experiments in series scheme
8.5. Convergence of the adapted statistical experiments
8.6. Scaling parameter estimation
8.7. Statistical estimations of the renewal intensity parameter
Chapter 9. Filtering of Stationary Gaussian Statistical Experiments
9.1. Stationary statistical experiments
9.2. Filtering of discrete Markov diffusion
9.3. The filtering error
9.4. The filtering empirical estimation
Chapter 10. Asymptotic Large Deviations for Markov Random Evolutionary Process
10.1. Asymptotic large deviations
10.2. Asymptotically stopped Markov random evolutionary process
10.3. Explicit representation for the normalizing function
Chapter 11. Asymptotic Large Deviations for Semi-Markov Random Evolutionary Processes
11.1. Recurrent semi-Markov random evolutionary processes
11.2. Asymptotic large deviations
Chapter 12. Heuristic Principles of Phase Merging in Reliability Analysis
12.1. The duplicated renewal system
12.2. The duplicated renewal system in the series scheme
12.3. Heuristic principles of the phase merging
12.4. The duplicated renewal system without failure
References
Index
Other titles from ISTE in Mathematics and Statistics
End User License Agreement
Guide
Cover
Table of Contents
Dedication Page
Title Page
Copyright Page
Preface
Introduction
Begin Reading
References
Index
Other titles from ISTE in Mathematics and Statistics
End User License Agreement
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“Nos esse quasi nanos gigantium humeris insidentes, ut possimus plura eis et remotiora videre,non utique proprii visus acumine, aut eminentia corporis, sed quia in altum subvehimur et extollimur magnitudine gigantea”(dicebat Bernardus Carnotensis)
In memory of our Teacher and Mentor, Vladimir Semenovich Koroliuk
Series EditorNikolaos Limnios
Asymptotic and Analytic Methods in Stochastic Evolutionary Systems
Dmitri KorolioukIgor Samoilenko
First published 2023 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2023The rights of Dmitri Koroliouk and Igor Samoilenko to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2023937508
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-911-2
Preface
This book is devoted to the study of Markov and semi-Markov random evolutionary processes.
The material presented in this book is classified by the authorship as follows:
–
Chapters 1
–
5
were written by Igor Samoilenko;
–
Chapters 6
–
12
were written by Dmitri Koroliouk.
This book is devoted to analytical methods for studying Markov and semi-Markov random evolutionary processes, in particular the derivation and solving of corresponding differential and integral equations, constructing new models, like fading evolution, etc.
The statistics of random evolutionary processes considers the problem of estimating parameters, as well as optimal filtering of stationary Gaussian evolutions.
Particular attention is paid to the problems of asymptotically large deviations for Markov and semi-Markov random evolutionary processes, in the sense of the random time of entering a region with the asymptotically decreasing probability of hitting it.
Dmitri Koroliouk expresses his appreciation and gratitude to the mathematical department of the University of Rome II – Tor Vergata, in particular to Professor Filippo Bracci for partially supporting the contents of this book.
Dmitri KOROLIOUKNTUU Igor Sikorsky Kyiv Polytechnic InstituteInstitute of Mathematics NASInstitute of Telecommunications and Global Information Space NASKyivUkraineandUniversity of Rome II – Tor VergataUNESCO Interdisciplinary Chair in Biotechnology and BioethicsRomeItaly
Igor SAMOILENKOTaras Shevchenko National University of KyivInstitute for Applied System Analysis ofNTUU Igor Sikorsky Kyiv Polytechnic InstituteKyivUkraine
March 2023
Introduction
This book continues with the studies found in our previous book Random Evolutionary Systems: Asymptotic Properties and Large Deviations in the Mathematics and Statistics series, edited by Nikolaos Limnios.
Here, we study some important aspects of random evolutionary systems, considering typical problems on some specific examples.
Numerous monographs have been devoted to the study and description of the Wiener process, in particular the fundamental works of Hida (1980) and Knight (1981). But such a process does not in fact exist in nature. In particular, Brownian motion is not a Wiener process, because the latter is only a mathematical idealization of physical Brownian motion. It is sufficient to indicate that when the property of continuous trajectories of the Wiener process is indifferentiable, a particle has no free run between two successive collisions; otherwise, it would have an infinite number of collisions per unit of time, which is unrealistic. Here, we examine models in which a particle has some free distance between two consecutive collisions. At the same time, we investigate two cases: Markov evolution, when the time during which the particle moves towards some direction is distributed exponentially with the parameter λ; semi-Markov evolution with an arbitrary distribution of the switching process. A similar example can be given by considering the process of heat distribution from a source in some environment. It is known that such a process in mathematical physics is described by a parabolic thermal conductivity equation, where parameters depend on the properties of the environment, and the initial conditions are determined by the initial position of the source or the initial distribution of heat in the medium. The classical theory proposes a model where the propagation of heat from the source occurs instantaneously, with an arbitrarily high speed, which is an analogue of the Wiener process. It is not difficult to note the analogy between the process of heat propagation, as a process of heat transfer by particles, and the process of Brownian motion. In both cases, we have diffusion with an infinite propagation speed. What distinguishes natural physical processes from their mathematical idealizations, which are the Wiener process and the classical model of heat conduction, is precisely the finiteness of the propagation speed. Many authors have studied various physical aspects of such models. The models investigated here describe the motion of a particle with a finite speed. Thus, the proposed random evolutionary process has the characteristics of a natural physical process: free run and finiteness propagation speed. In the proposed models, the number of possible directions of evolution can be both finite and infinite. In the theory of random evolutions (TRE), an asymptotic technique has been developed, which is based on the well-developed apparatus of the theory of random products of matrices and operators (Furstenberg 1963; Grenander 1963; Tutubalin 1965; Hannan 1965) and the theory of singularly perturbed semigroups of operators. The main goal of these studies is the proof of various limit theorems. In this aspect, it is necessary to mention the works of American mathematicians Griego and Hersh (1969, 1971), Papanicolaou (1971a), Papanicolaou and Kellek (1971), Gorostiza (1972, 1973b), Hersh and Pinsky (1972), Hersh (1974), Pinsky (1991) and others. An effective means of proving limit theorems in TRE is the theory of phase merging of complex systems developed by Korolyuk and Turbin (1993) and by Korolyuk and Limnios (2005). Thanks to their methods, many authors have proven important limit theorems on regular approximations to solutions of singularly perturbed differential equations for various stochastic models, as well as for dynamic systems with coefficients dependent on Markov processes, for example, Skorokhod (1989). Korolyuk and Swishchuk (1995a, 1995b) developed the theory of semi-Markov random evolutions based on the theory of martingales. But at the same time, it is not only the limit theorems which lead to Wiener or diffusion processes which are of interest, but also the equation and systems of equations describing the limit behavior of one or another characteristics of a random evolutionary system. Thanks to this, it is possible to get various limit theorems, based on the equation by the limit transition. This work is dedicated to this topic. The boundary behavior of the Markov random evolutionary process was studied, equations describing this process were found and methods for solving these equations were proposed. It should be noted that the first work in this particular field was the work of Goldstein (1951). By studying the simplest model – one-dimensional random evolution governed by a Poisson process – he obtained the equation
which is known as a damped wave or telegraph equation. In this case, the constants v and λ refer to the speed of the particle and the intensity of the switching process in accordance. Goldstein proved that the density of the probability distribution of the coordinate of a particle on a straight line satisfies the equation found and, with the correspondingly given initial conditions, a solution to the Cauchy problem can also be found. Further research in this direction was carried out by Kac, who in 1956 showed that not only the density distribution, but also any measurable functional of the particle trajectory (Kac 1957, 1974) satisfy this equation. He also found the solution of the Cauchy problem in the form of expectation functions from a random process and proved the validity of such a formula for more general equations. Here, we describe some generalizations of the Goldstein–Katz model to the case of the space Rn. Kac was also the first to notice that if v → ∞ and λ → ∞ so that then such a model is an asymptotically Wiener process with zero drift and the diffusion coefficient σ2. We have proven the theorem about weak convergence of measures generated by a Markov random evolutionary process in Rn to the measure of a Wiener process. Subsequently, the Goldstein–Katz model was generalized to the case of the different intensity of the controlling Poisson process when moving in the positive and negative directions. Bartlett (1957, 1978), Kaplan (1964), Cane (1967, 1975), Griego and Hersh (1969, 1971) generalized probabilistic Kac’s formula for the solution of the abstract telegraph equation where a > 0 is some constant, and A is the infinitesimal operator of the semigroup of compressions of the class C0 of operators in some Banach space. Kisinski (1974) and other authors further strengthened this result by generalizing Kac’s probabilistic formula to the solution of the equation is any integer.
It is also necessary to mention numerous works of the Italian mathematician Orsingher that were devoted to the equations arising in various one-dimensional model random evolutions (Orsingher 1986; Orsingher and Sommella 2004), some models on the plane (Orsingher 1985, 1990b; Orsingher and Kolesnik 1994) and on the sphere (Orsingher 1987c), etc., as well as their solution and physical interpretation of solutions. However, it should be noted that the works of Orsingher are mostly devoted to the study of processes described by equations that can be factored. The solutions of such equations are written down as a linear combination of known solutions of factorization elements. We get equations that are not factorable, and two solution methods are proposed for them.
In particular, when studying the Markov random evolutionary process, integral equations are applied, which makes it possible to describe processes governed by a non-exponential distribution (semi-Markov evolutionary process), as well as write down equations for processes for which it is impossible to find differential equations directly (fading evolutionary process).
Other works on this topic should also be noted: Jenssen (1990), Pinsky (1991), Foong (1992) and Foong and Kanno (1994).
In Chapter 1, we define multidimensional models of the Kac type, describe corresponding moments, systems of Kolmogorov equations and some results on their convergence of the Wiener process.
Chapter 2 describes some symmetric properties of the Markov random evolutionary processes in Rn and implies some generalizations of well-known symmetric distributions.
Chapter 3 is devoted to the description of hyperparabolic equations and the integral equation for Markov random evolutionary processes in Rn, methods of their solutions for specific initial conditions. An obvious view of the corresponding distribution is also presented.
Chapter 4 deals with a special case, model of the fading Markov random evolutionary process, which cannot be studied with the use of differential equations only. At the same time, it is interesting because of the existence of the limit distribution at t → ∞, which is impossible in the classical model.
Chapter 5 presents two models of evolutionary processes, namely the evolution of Kac’s type on a complex plane, in which moments define solutions of the Cauchy problem for a Schrödinger-type equation, and the evolutionary process in Rn with infinitely many directions, for which methods of our previous book Random Evolutionary Systems: Asymptotic Properties and Large Deviations are applied to obtain the limit process in the sense of the Kac-type hydrodynamic limit.
In Chapter 6, we consider a random evolution ζ(t), t 0, which depends on a random environment Y (t), t 0, which, in turn, is switched by an embedded Markov chain Xk, k 0, considering a specific relation between the continuous-time t 0 and the discrete-time k 0. The purpose is to prove the convergence (in distribution) of the process ζ(t), t 0 to the Ornstein–Uhlenbeck process under some scaling of the process and its time parameter. The limit will be considered by a small series parameter ε > 0, ε → 0.
Chapter 7 is devoted to the filtration of random Gaussian–Markov evolutionary processes represented by stationary Gaussian statistical experiments, determined by the solution of the optimal filtration equation. It is characterized by a two-dimensional matrix of covariances. The parameters of the filtered signal are set by empirical covariances.
In Chapter 8, we study a random evolutionary process, represented as statistical experiments with the random change of time, which transforms a discrete stochastic basis in a continuous one. The adapted stochastic experiments are studied in the continuous stochastic basis in the series scheme. The transition to limit by the series parameter generates an approximation of adapted statistical experiments by a diffusion process with evolution.
The average intensity parameter of renewal times is estimated in three different cases: the Poisson renewal process, a stationary renewal process with delay and the general renewal process with the Weibull–Gnedenko renewal time distribution.
Chapter 9 repeats the statement of the filtering problem discussed in Chapter 7. However, the analysis of this problem differs significantly from the previous one. This chapter is placed separately from Chapter 7 intentionally, in order to separate the analytical approach to the problem. The performed analysis casts doubt on the well-known fact about the equivalence of the independence and non-correlation conditions for Gaussian random sequences and processes, despite the indisputable proof of the equivalence theorem.
In Chapter 10, we discuss the necessary and sufficient conditions for the convergence of the distributions of normalized first entry times into asymptotically receding domains for asymptotic large deviations for Markov random evolutionary processes.
In Chapter 11, we discuss the necessary and sufficient conditions for the convergence of the distributions of normalized first entry times into asymptotically large deviations for semi-Markov random evolutionary processes.
Chapter 12 deals with some analytical methods in reliability theory that stimulate the consideration of the principles of merging subsets of states (phases) for Markov and semi-Markov processes. The peculiarity of the algorithm is that the heuristic principles of phase merging can be constructively used to analyze the reliability of stochastic systems. In particular, the stationary phase merging algorithm can be used to simplify the analysis of the reliability of a redundant recovery system.
