Markov Processes with a Large Number of Locally Interacting Components: Existence of a Limit Process and Its Ergodicity

Markov processes with continuous time are considered. Elements of the phase space of these processes constitute a collection of a large number of components, each of which assumes a discrete series of values. It is assumed that these components are compared with points of a discrete grid in space and that the statistical relationships of individual component variations are determined by the states of the neighboring components. Such processes appear in various cybernetic problems. Their investigation is carried out with the aid of a transition to a limiting case, when the number of components is infinite.