Markov Random Fields

In this book we study Markov random functions of several variables. What is traditionally meant by the Markov property for a random process (a random function of one time variable) is connected to the concept of the phase state of the process and refers to the independence of the behavior of the process in the future from its behavior in the past, given knowledge of its state at the present moment. Extension to a generalized random process immediately raises nontrivial questions about the definition of a suitable" phase state," so that given the state, future behavior does not depend on past behavior. Attempts to translate the Markov property to random functions of multi-dimensional "time," where the role of "past" and "future" are taken by arbitrary complementary regions in an appro­ priate multi-dimensional time domain have, until comparatively recently, been carried out only in the framework of isolated examples. How the Markov property should be formulated for generalized random functions of several variables is the principal question in this book. We think that it has been substantially answered by recent results establishing the Markov property for a whole collection of different classes of random functions. These results are interesting for their applications as well as for the theory. In establishing them, we found it useful to introduce a general probability model which we have called a random field. In this book we investigate random fields on continuous time domains. Contents CHAPTER 1 General Facts About Probability Distributions §1.

1 General Facts About Probability Distributions.- §1. Probability Spaces.- 1. Measurable Spaces.- 2. Distributions and Measures.- 3. Probability Spaces.- §2. Conditional Distributions.- 1. Conditional Expectation.- 2. Conditional Probability Distributions.- §3. Zero-One Laws. Regularity.- 1. Zero-One Law.- 2. Decomposition Into Regular Components.- §4. 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.- §5. 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.- §1. 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.- §2. Stopping ?-algebras. Random Sets and the Strong Markov Property.- 1. Stopping ?-algebras.- 2. Random Sets.- 3. Compatible Random Sets.- 4. Strong Markov Property.- §3. 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.- §1. 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.- §2. Stationary Generalized Functions.- 1. Spectral Representation of Coupled Stationary Generalized Functions.- 2. Biorthogonal Stationary Functions.- 3. The Duality Condition and a Markov Criterion.- §3. Biorthogonal Generalized Functions Given by a Differential Form.- 1. Basic Definitions.- 2. Conditions for Markov Behavior.- §4. 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.- §5. Stochastic Differential Equations.- 1. Markov Transformations of “White Noise”.- 2. The Interpolation and Extrapolation Problems.- 3. The Brownian Sheet.- 4 Vector-Valued Stationary Functions.- §1. Conditions for Existence of the Dual Field.- 1. Spectral Properties.- 2. Duality.- §2. The Markov Property for Stationary Functions.- 1. The Markov Property When a Dual Field Exists.- 2. Analytic Markov Conditions.- §3. Markov Extensions of Random Processes.- 1. Minimal Nonanticipating Extension.- 2. Markov Stationary Processes.- 3. Stationary Processes with Symmetric Spectra.- Notes.