Markov Random Fields

1 General Facts About Probability Distributions.- x1. Probability Spaces.- 1. Measurable Spaces.- 2. Distributions and Measures.- 3. Probability Spaces.- x2. Conditional Distributions.- 1. Conditional Expectation.- 2. Conditional Probability Distributions.- x3. Zero-One Laws. Regularity.- 1. Zero-One Law.- 2. Decomposition Into Regular Components.- x4. Consistent Conditional Distributions.- 1. Consistent Conditional Distributions for a Given Probability Measure.- 2. Probability Measures with Given Conditional Distributions.- 3. Construction of Consistent Conditional Distributions.- x5. Gaussian Probability Distributions.- 1. Basic Definitions and Examples.- 2. Some Useful Propositions.- 3. Gaussian Linear Functionals on Countably-Normed Hilbert Spaces.- 4. Polynomials of Gaussian Variables and Their Conditional Expectations.- 5. Hermite Polynomials and Multiple Stochastic Integrals.- 2 Markov Random Fields.- x1. Basic Definitions and Useful Propositions.- 1. Splitting ?-algebras.- 2. Markov Random Processes.- 3. Random Fields; Markov Property.- 4. Transformations of Distributions which Preserve the Markov Property. Additive Functionals.- x2. Stopping ?-algebras. Random Sets and the Strong Markov Property.- 1. Stopping ?-algebras.- 2. Random Sets.- 3. Compatible Random Sets.- 4. Strong Markov Property.- x3. Gaussian Fields. Markov Behavior in the Wide Sense.- 1. Gaussian Random Fields.- 2. Splitting Spaces.- 3. Markov Property.- 4. Orthogonal Random Fields.- 5. Dual Fields. A Markov Criterion.- 6. Regularity Condition. Decomposition of a Markov Field into Regular and Singular Components.- 3 The Markov Property for Generalized Random Functions.- x1. Biorthogonal Generalized Functions and the Duality Property.- 1. The Meaning of Biorthogonality for Generalized Functions in Hilbert Space.- 2. Duality of Biorthogonal Functions.- 3. The Markov Property for Generalized Functions.- x2. Stationary Generalized Functions.- 1. Spectral Representation of Coupled Stationary Generalized Functions.- 2. Biorthogonal Stationary Functions.- 3. The Duality Condition and a Markov Criterion.- x3. Biorthogonal Generalized Functions Given by a Differential Form.- 1. Basic Definitions.- 2. Conditions for Markov Behavior.- x4. Markov Random Functions Generated by Elliptic Differential Forms.- 1. Levy Brownian Motion.- 2. Structure of Spaces for Given Elliptic Forms.- 3. Boundary Conditions.- 4. Regularity and the Dirichlet Problem.- x5. Stochastic Differential Equations.- 1. Markov Transformations of "White Noise".- 2. The Interpolation and Extrapolation Problems.- 3. The Brownian Sheet.- 4 Vector-Valued Stationary Functions.- x1. Conditions for Existence of the Dual Field.- 1. Spectral Properties.- 2. Duality.- x2. The Markov Property for Stationary Functions.- 1. The Markov Property When a Dual Field Exists.- 2. Analytic Markov Conditions.- x3. Markov Extensions of Random Processes.- 1. Minimal Nonanticipating Extension.- 2. Markov Stationary Processes.- 3. Stationary Processes with Symmetric Spectra.- Notes.