Mathematics of Random Phenomena

A)Theory.- I. Mathematical Prerequisites.- I.1. Linear Algebra.- I.2. Transposition.- I.3. Functional Analysis.- I.4. Prerequisites from the Theory of Measure and Integration.- I.5. Sobolev Spaces.- II. Principles of Probability Theory.- II.1. Principles of Probability Theory.- II.2. Characteristic Functions. Moments.- II.3. Common Probability Laws.- II.4. Relation Between Probability Theory and Statistics.- II.5. Effective Description of Probability Laws.- III. Stochastic Processes and Random Fields.- III.1. Classical Stochastic Processes.- III.2. Processes of Second Order.- III.3. Transformations of Stochastic Processes.- III.4. Integral Representation of Classical Stochastic Processes of Second Order.- III.5. Conditional Expectations and Applications to Processes.- III.6. The Markovian Property.- III.7. Statistics on Random Functions.- IV. Linear Filtering and Spectral Analysis.- IV.1. Linear Transformations of Signals.- IV.2. Convolution Filters.- IV.3. Finite-Dimensional Linear Oscillators.- IV.4. Linear Integral Transformations of M.S. Continuous Second-Order Stochastic Processes.- IV.5. Convolution Filtering of M.S. Continuous Second-Order Stationary Processes.- IV.6. Stochastic Linear Oscillators.- V. Statistics on Sample Paths.- V.1. Introduction.- V.2. Processes with Absolutely Continuous Sample Paths.- V.3. Polygonal Approximation of Sample Paths.- V.4. Mean Number of Crossings for Almost All Levels.- V.5. Mean Number of Crossings.- V.6. Mean Number of Up crossings (Downcrossings).- V.7. Relationship Between the Mean Number of Upcrossings and the Study of Absolute Maxima on an Interval.- V.8. Mean Number of Local Maxima and Minima.- B)Applications.- VI. Dynamic Response of Structures Subjected to Wind Loads.- VI.1. Brief Review of Fluid Mechanics.- VI.2. Probabilistic Description of the Atmospheric Turbulent Boundary Layer.- VI.3. Along-Wind Dynamic Response to Flexible Structures.- VI.4. Experimental Verification of the Model>.- VII. Probabilistic Approach to the Effects of Earthquakes on Structures.- VII.1. General Facts.- VII.2. Probabilistic Modelling of Strong-Motion Earthquakes.- VII.3. Responses of Systems with Linear Behavior.- VII.4. Response of Structures with Nonlinear Behavior.- VII.5. Numerical Resolution of the Fokker-Planck Equation.- VIII. Probabilistics Models of Sea Wave and Applications.- VIII.1. Statement of the Problem and Hypotheses.- VIII.2. Prerequisites on the Deterministic Wave.- VIII.3. Probabilistic Model of Short-Term Sea States.- VIII.4. Application to the Study of the Response of a Certain Type of Offshore Structures.- IX. Mechanical Fatigue under Random Loading.- IX.1. General Facts Concerning Fatigue.- IX.2. Various Notions of Cycles. Counting Methods.- IX.3. Fatigue under Constant-Cycle Loading.- IX.4. Fatigue under Loading with Constant Cycles on Intervals.- IX.5.Fatigue under Random Loading.- C)Applications.- X. Measures on Vector Spaces and Stochastic Modelling.- X.1. Locally Convex Spaces.- X.2. Duality for Locally Convex Spaces.- X.3. Cylindrical Models and Models of Random Elements.- X.4. Study of Certain ?-Algebras.- X.5. Constructing Models.- X.6. Linear Processes of Order p..- X.7. Stochastic Hilbert Space. Reproducing Hilbert Space.- X.8. Weak Integrals.- XI. Generalized Stochastic Processes.- XI.1. Random Vector Distributions.- XI.2. Linear Transformations of Generalized Processes.- XI.3. Spectral Theory of Generalized Processes.- XI.4. Linear Equations.- XI.5. Linear Oscillators with Arbitrarily Many Degrees of Freedom.- XI.6. Stochastic Hilbert Spaces and Regularity of Generalized Processes.- XII. Markov Realization of Gaussian Processes.- XII.1. Physically Realizable Processes.- XII.2. Markov Realization of Processes with Spectral Density of Rational Type.- XII.3. Approximate Markov Realization of Physically Realizable Processes.- XII.4. Filtering Techniques for Random Fields.- XII.5. Application of Random Field Filtering to Approximate Markovianization.- XIII. Stochastic Differential Equations.- XIII.1. Stochastic Integrals.- XIII.2. Indefinite Stochastic Integrals and Ito's Differential Calculus.- XIII.3. Existence and Uniqueness of Solutions in the Uniform Lipschitz Case.- XIII.4. Existence and Uniqueness of Solutions in other Cases.- XIII.5. The Fokker-Planck Equation.- XIII.6. Law of the First Exit from an Open Set.- XIII.7. Markovianization of Some Stochastic Initial-Value Problems.- XIV. Random Oscillations with Intermittent Constraints.- XIV.1. Deterministic Multivalued Differential Equations.- XIV.2. Deterministic Oscillations with Intermittent Constraints.- XIV.3. Multivalued Stochastic Differential Equations.- XIV.4. The Diffusion Operator of (MSDE).- XIV.5. Diffusion Equation for the Probability Distributions for (MSDE) in the Product Situation.- XIV.6. Multivalued Equations and Stochastic Differential Equations with Boundary Conditions.- XIV.7. Generalized FPE for a (MSDE) of Product type.- XV. Linear Oscillators Subjected to Squared Gaussian Processes.- XV.1. Existence Theorem for Reproducing Hilbert Spaces.- XV.2. Fourier Transform of Images of Gaussian Measures under Certain Nonlinear Mappings.- XV.3. Proof of the Topological Property XV.2.4.c.- XV.4. Example of a Nonlinear Transformation of a Gaussian Measure.- XV.5. An Approximation Procedure.- Additional References.