Locally interacting processes with a noncompact set of values (Q1103977)

Markov chains of fields \(\xi_ t:\) \(Z^{\nu}\to R^ k\), are considered, where the probability distribution of \(\xi_{t+1}(y)\) depends on \(\xi_ t(x)\) for finitely many x's only (local interaction). Theorem 1 states the exponential convergence of the distributions of \(\xi_ t\) in some suitable metric for a recursively described chain. Theorem 2 concerns \(\xi_ t\) which is described with the help of independent Markov chains on U, where \(\xi_ t:\) \(Z^{\nu}\to U\). The transition probabilities \(P(\xi_{t+1}(x)=u| \xi_ t)\) are defined by these Markov chains and include some interaction between x and finitely many y's from \(Z^{\nu}\) multiplied by a small parameter \(\delta\). The assertion is that the distributions of \(\xi_ t\) converge in variation and the finite-dimensional limit distributions depend analytically on the small parameter \(\delta\). The proof of Theorem 2 is not given. It is only indicated that the proof uses techniques of cluster expansion and Lyapunov functions studied by the first author with \textit{M. V. Men'shikov} [Tr. Mosk. Mat. O. - va 39, 3-48 (1979; Zbl 0441.60073)], and with \textit{R. A. Minlos} [Stochastic Gibbs fields. The method of cluster expansions (1985; Zbl 0584.60062)], respectively.