High Dimensional Nonlinear Diffusion Stochastic Processes

Preface......Page 8
Contents......Page 14
1.1 Prerequisites for Reading......Page 20
1.2 Random Variable. Stochastic Process. Random Field. High-Dimensional Process. One-Point Process......Page 22
1.3 Two-Point Process. Expectation. Markov Process. Example of Non-Markov Process Associated with Multidimensional Markov Process......Page 29
1.4 Preceding Subsequent and Transition Probability Densities. The Chapman-Kolmogorov Equation. Initial Condition for Markov Process......Page 35
1.4.1 The Chapman-Kolmogorov equation......Page 38
1.4.2 Initial condition for Markov process......Page 39
1.5 Homogeneous Markov Process. Example of Markov Process: The Wiener Process......Page 42
1.6 Expectation Variance and Standard Deviations of Markov Process......Page 45
1.7 Invariant and Stationary Markov Processes. Covariance. Spectral Densities......Page 49
1.8 Diffusion Process......Page 56
1.9 Example of Diffusion Processes: Solutions of Ito's Stochastic Ordinary Differential Equation......Page 59
1.10 The Kolmogorov Backward Equation......Page 65
1.11 Figures of Merit. Diffusion Modelling of High-Dimensional Systems......Page 67
1.12.1 Probability density......Page 70
1.12.2 Invariant probability density......Page 73
1.12.3 Stationary probability density......Page 76
1.13 The Purpose and Content of This Book......Page 79
2.1 Introduction......Page 82
2.2 Time-Derivatives of Expectation and Variance......Page 83
2.3.1 The first-order system......Page 85
2.3.2 The second-order system......Page 87
2.3.3 Systems of the higher orders......Page 89
2.4.1 The case of stochastic resonance......Page 90
2.4.2 Practically efficient implementation of the second-order system......Page 92
2.5.1 Damping matrix......Page 95
2.5.2 The uncorrelated-matrixes approximation......Page 96
2.5.3 Nonlinearity of the drift function......Page 99
2.5.4 Fundamental limitation of the state-space-independent approximations for the diffusion and damping matrixes......Page 100
2.6 The Steady-State Approximation for The Probability Density......Page 101
3.2 Preliminary Remarks......Page 104
3.3 Expectation. The Finite-Equation Method......Page 105
3.4 Explicit Expression for Variance......Page 107
3.5.1 Partial differential equation for logarithm of the density......Page 109
3.5.2 Truncated equation for the logarithm and the detailed-balance equation......Page 110
3.5.3 Case of the detailed balance......Page 112
3.5.4 The detailed-balance approximation......Page 114
3.5.5 The simplified detailed-balance approximation. Theorem on the approximating density......Page 115
3.6 Analytical-Numerical Approach to Non-Invariant and Invariant Diffusion Processes......Page 118
3.6.1 Choice of the bounded domain of the integration......Page 119
3.6.2 Evaluation of the multifold integrals. The Monte Carlo technique......Page 121
3.6.3 Summary of the approach......Page 123
3.7 Discussion......Page 124
4.1 Introduction......Page 126
4.2 Previous Results Related to Covariance and Spectral-Density Matrixes......Page 127
4.3 Time-Separation Derivative of Covariance in the Limit Case of Zero Time Separation......Page 128
4.4 Flicker Effect......Page 130
4.5 Time-Separation Derivative of Covariance in the General Case......Page 131
4.6 Case of the Uncorrelated Matrixes......Page 133
4.7 Representations for Spectral Density in the Uncorrelated-Matrixes Case......Page 136
4.8 Example: Comparison of the Dampings for a Particle Near the Minimum of Its Potential Energy......Page 137
4.9 The Deterministic-Transition Approximation......Page 142
4.10.1 Covariance in the general case......Page 145
4.10.2 Covariance and the qualities related to it in a simple fluid......Page 147
4.10.3 Case of the hard-sphere fluid......Page 149
4.10.4 Summary of the procedure in the general case......Page 152
4.11.1 Practical issues......Page 153
4.11.2 Summary of the approach......Page 155
4.12 Discussion......Page 156
5.1 Introduction......Page 160
5.2 Various Types of Ito's Stochastic Differential Equations......Page 161
5.3 Method to Reduce ISPIDE to System of ISODEs......Page 163
5.3.1 Projection approach......Page 167
5.3.2 Stochastic collocation method......Page 170
5.3.3 Stochastic-adaptive-interpolation method......Page 172
5.4 Related Computational Issues......Page 178
5.5 Discussion......Page 179
6.1 Introduction......Page 182
6.2.1 Microscopic random walks in deterministic macroscopic models of multiparticle systems......Page 184
6.2.2 Macroscale mesoscale and microscale domains in terms of the wave-diffusion equation......Page 190
6.2.3 Stochastic generalization of the deterministic macroscopic models of multiparticle systems......Page 195
6.3.1 Deterministic model for electron fluid in semiconductor......Page 196
6.3.2 Mesoscopic wave-diffusion equations in the deterministic fluid-dynamic model......Page 199
6.3.3 Stochastic generalization of the deterministic fluid-dynamic model. The semiconductor-fluid ISPDE system......Page 206
6.4.1 The SF-ISPDE system in connection with semiconductor noises......Page 211
6.4.2 Some directions for future development......Page 214
7.1 High-Dimensional Diffusion Processes......Page 216
7.3 Reasonable Amount of the Main Computer Memory......Page 217
7.4 Real-Time Signal Transformation by Diffusion Process......Page 218
8.1 Analytical-Numerical Approach to Engineering Problems......Page 220
8.2 Severe Practical Limitations of Common Analytical Techniques in the High-Dimensional Case. Possible Alternatives......Page 222
Appendix A Example of Markov Processes: Solutions of the Cauchy Problems for Ordinary Differential Equation System......Page 224
Appendix B Signal-to-Noise Ratio......Page 228
C.1 Description of the Model......Page 232
C.2 Energy-Independent Momentum-Relaxation Time. Equilibrium Probability Density......Page 239
C.3 Energy-Dependent Momentum-Relaxation Time: General Case......Page 241
C.4 Energy-Dependent Momentum-Relaxation Time: Case of Simple Fluid......Page 246
D.1 Proof of Theorem 2.1......Page 250
D.2 Proof of Theorem 2.2......Page 251
D.3 Green's Formula for the Differential Operator of Kolmogorov's Backward Equation......Page 254
D.4 Proof of Theorem 2.3......Page 256
D.5 Quasi-Neutral Equilibrium Point......Page 258
D.6 Proof of Theorem 2.4......Page 260
E.2 Proof of Theorem 4.2......Page 262
E.3 Proof of Theorem 4.3......Page 263
E.4 Proof of Theorem 4.4......Page 264
Appendix F Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and Chemical-Reaction / Generation-Recombination Noise......Page 266
Appendix G Example: Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in Three-Dimensional Space......Page 274
Appendix H Resources for Engineering Parallel Computing under Windows 95......Page 280
Bibliography......Page 284
Index......Page 300