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