Markov properties of MM-class processes with two parameters (Q1898419)

Using the plot of growing single-point Markov processes on a single- parameter Markov process, it is constructed successfully a class of important two-parameter processes which are called MM-class processes and whose two parameters are unequal in status. It has been researched if MM- class processes possess the various two-parameter Markov properties. Let \((E, {\mathcal E})\) be a measurable space, all sets consisting of a single point belonging to \({\mathcal E}\), \(\delta (y)\) denotes the probability measure on \((E, {\mathcal E})\), which centralizes at a single point \(y \in E\). Let \((\Omega, {\mathcal F}, P)\) be a probability space, on which a lot of homogeneous Markov processes with a single-parameter is defined as follows: \(Y = \{Y(t), t \geq 0\}\), \(U_t^y = \{U^y_t (s), s \geq 0\}\), \(t \geq 0\), \(y \in E\). Their phase spaces are the same \((E, {\mathcal E})\). Assume that they are independent mutually and that for \(t \geq 0\), \(y \in E\), processes \(U^y_t\) possess the same homogeneous family of transition functions, \(\overline P = \{\overline P (s,y,B), s \geq 0, y \in E, B \in {\mathcal E}\}\), the initial measure of \(U^y_t\) is \( \delta (y)\). If \(X(s,t) = U_t^{Y(t)} (s)\) for \(s \geq 0\), \(t \geq 0\), the two-parameter process \(X = \{X (s,t), s \geq 0, t \geq 0\}\) is called an MM-class process. If the initial measure of process \(Y\) is denoted by \(\widehat q\) and the family of homogeneous transition functions of process \(Y\) is denoted by \(\widehat P = \{\widehat P (t,y,B), t \geq 0, y \in E, B \in {\mathcal E}\}\), the process \(X\) is called \((\widehat q, \widehat P, \overline P)\)-MM-class process. It follows that: 1. Let \((E, {\mathcal E})\) be a \(\sigma\)-compact metric measurable space, given a probability measure \(\widehat q\) on \((E, {\mathcal E})\) and two families of homogeneous transition functions, \(\widehat P\) and \(\overline P\). Then there exists a \((\widehat q, \widehat P, \overline P)\)-MM-class process \(X\) defined on some probability space. 2. An MM-class process possesses the wide-past Markov property and 1-Markov property, and has the family of three-point transition functions which are \(P(s,t,u,v; a,b,c;B) = \overline P (u,b,B)\), \(s,t,u,v \geq 0\), \(a,b,c \in E\), \(B \in {\mathcal E}\). 3. A \((\widehat q, \widehat P, \overline P)\)-MM-class process possesses \(*\)-Markov property, single-point Markov property, wide-future Markov property, and 2-Markov property if and only if \(\overline P\) is degenerate, i.e., \(\overline P (s,a, \cdot) = \delta\)(a) for \(s \geq 0\) and \(a \in E\).