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 λ; 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.
Table of Contents
Preface ix
Introduction xi
Chapter 1 Multidimensional Models of Kac Type 1
1.1. Definitions and basic properties 1
1.2. Moments of evolutionary process 8
1.3. Systems of Kolmogorov equations 17
1.4. Evolutionary operator and theorem about weak convergence to the measure of the Wiener process 23
Chapter 2 Symmetry of Markov Random Evolutionary Processes in Rn 29
2.1. Symmetrization: definition and properties 29
2.2. Examples of symmetric distributions in Rn and distributions on n + 1-hedra32
2.2.1. Symmetric distributions 32
2.2.2. Distributions on n + 1-hedra 35
Chapter 3 Hyperparabolic Equations, Integral Equation and Distribution for Markov Random Evolutionary Processes 39
3.1. Hyperparabolic equations and methods of solving Cauchy problems 39
3.2. Analytical solution of a hyperparabolic equation with real-analytic initial conditions 46
3.3. Integral representation of the hyperparabolic equation 57
3.4. Distribution function of evolutionary process 67
Chapter 4 Fading Markov Random Evolutionary Process 77
4.1. Definition of fading Markov random evolutionary process, its moments and limit distribution 77
4.2. Integral equation for a function from the fading random evolutionary process 89
4.3. Equations in partial derivatives for a function of the fading random evolutionary process 93
Chapter 5 Two Models of the Evolutionary Process 99
5.1. Evolution on a complex plane 99
5.2. Evolution with infinitely many directions 109
5.2.1. Symmetric case 110
5.2.2. Non-symmetric case 119
Chapter 6 Diffusion Process with Evolution and Its Parameter Estimation 125
6.1. Asymptotic diffusion environment 125
6.2. Approximation of a discrete Markov process in asymptotic diffusion environment 127
6.3. Parameter estimation of the limit process 132
Chapter 7 Filtration of Stationary Gaussian Statistical Experiments 135
7.1. Introduction 135
7.2. Stochastic difference equation of the process of filtration 137
7.3. Coefficient of filtration 138
7.4. Equation of optimal filtration 139
7.5. Characterization of a filtered signal 141
Chapter 8 Adapted Statistical Experiments with Random Change of Time 143
8.1. Introduction 143
8.2. Statistical experiments and evolutionary processes 144
8.3. Stochastic dynamics of statistical experiments 145
8.4. Adapted statistical experiments in series scheme 147
8.5. Convergence of the adapted statistical experiments 149
8.6. Scaling parameter estimation 154
8.7. Statistical estimations of the renewal intensity parameter 155
8.7.1. Poisson’s renewal process with parameter q =2 156
8.7.2. Stationary renewal process with delay, determined by the initial distribution function of the limit over jumps 156
8.7.3. Renewal processes with arbitrarily distributed renewal intervals 157
Chapter 9 Filtering of Stationary Gaussian Statistical Experiments 159
9.1. Stationary statistical experiments 159
9.2. Filtering of discrete Markov diffusion 161
9.3. The filtering error 164
9.4. The filtering empirical estimation 166
Chapter 10 Asymptotic Large Deviations for Markov Random Evolutionary Process 171
10.1. Asymptotic large deviations 171
10.2. Asymptotically stopped Markov random evolutionary process 191
10.3. Explicit representation for the normalizing function 206
Chapter 11 Asymptotic Large Deviations for Semi-Markov Random Evolutionary Processes 209
11.1. Recurrent semi-Markov random evolutionary processes 209
11.2. Asymptotic large deviations 212
Chapter 12 Heuristic Principles of Phase Merging in Reliability Analysis 221
12.1. The duplicated renewal system 221
12.2. The duplicated renewal system in the series scheme 222
12.3. Heuristic principles of the phase merging 223
12.4. The duplicated renewal system without failure 225
References 227
Index 233
