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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.



Chapter 1. General Facts About Probability Distributions

Let X be an arbitrary set. When we consider elements xX and sets AX, we call X a space.
Yu. A. Rozanov

Chapter 2. Markov Random Fields

Let A1, , A2 be σ-algebras of events having the following relationship: if the outcomes of all events in are known, events A2A2 are independent of events A1A1. More precisely, the σ-algebras A1 and A2 are conditionally independent with respect to ; this gives the equation for conditional probabilities:
$$ P({A_1} \cdot {A_2}|B) = P({A_1}|B) \cdot P({A_2}|B) $$
for any A1A1, A2A2. We say that the σ-algebra ℬ splits A1 and A2 (or is splitting) if (1.1) holds for A1, , A2.
Yu. A. Rozanov

Chapter 3. The Markov Property for Generalized Random Functions

We have previously stipulated that by a generalized random function on a domain T ⊆ ℝ d we mean a continuous linear mapping of C 0 (T), the space of infinitely differentiable functions u = u(t), tT,into the Hilbert space L2(Ω, A, P).
Yu. A. Rozanov

Chapter 4. Vector-Valued Stationary Functions

We will consider stationary random functions ξ(t), t ∈ ℝ d , whose values are random elements (vectors) in a Hilbert space X with norm \( ||\xi (t)|| \)
$$ E||\zeta (t)|{|^2} < \infty ,t \in {\mathbb{R}^d} $$
and also random fields formed by the spaces H(S), S ⊆ ℝ d ,each of which is the closed linear span of the variables in the space L2(Ω, A, P)
$$ {\xi _x}(t) = (\xi (t),x) $$
defined by the scalar product in X of ξ(t),tS, and x ∈ X. Stationarity will be understood in the following sense:
$$ {\xi _x}(t + s) = {U_s}{\xi _x}(t),t,s \in {\mathbb{R}^d} $$
for all x ∈ X, where U s , s ∈ ℝd, form a continuous group of unitary operators in the space H(ℝd), usually called the shift operators.
Yu. A. Rozanov


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