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 approยญ 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 ยง1
CONTENT
1 General Facts About Probability Distributions -- ยง1. Probability Spaces -- ยง2. Conditional Distributions -- ยง3. Zero-One Laws. Regularity -- ยง4. Consistent Conditional Distributions -- ยง5. Gaussian Probability Distributions -- 2 Markov Random Fields -- ยง1. Basic Definitions and Useful Propositions -- ยง2. Stopping ?-algebras. Random Sets and the Strong Markov Property -- ยง3. Gaussian Fields. Markov Behavior in the Wide Sense -- 3 The Markov Property for Generalized Random Functions -- ยง1. Biorthogonal Generalized Functions and the Duality Property -- ยง2. Stationary Generalized Functions -- ยง3. Biorthogonal Generalized Functions Given by a Differential Form -- ยง4. Markov Random Functions Generated by Elliptic Differential Forms -- ยง5. Stochastic Differential Equations -- 4 Vector-Valued Stationary Functions -- ยง1. Conditions for Existence of the Dual Field -- ยง2. The Markov Property for Stationary Functions -- ยง3. Markov Extensions of Random Processes -- Notes
