A coupling of infinite particle systems
The Markov processes generated by interacted particle systems evoluting on one-dimensional lattice \(\mathbb{Z}\) are considered. The site state spaces for each system are isomorphic to each other and are the copies of a finite set \(\mathbb{D}\). Thus, the processes are Markov chains with continuous time and with a countable set of states. The generator \(\Omega\) of such a process is written in the form \[ (\Omega f) (\eta) = \sum_T \sum_{\nu \in \mathbb{D}^T} \bigl( f(\nu \eta) - f(\eta) \bigr) c_T (\eta, \nu), \] where \(f(\eta)\) is a distribution function depending on the configurations \(\eta (y)\), \(y \in \mathbb{Z}\), \(\eta (y) \in \mathbb{D}\), the first sum is over finite subsets \(T\) of \(\mathbb{Z}\) and where \(\nu \eta\) denotes the configuration with \(\nu \eta\) equal to \(\nu (y)\) if \(y \in T\) and equal to \(\eta (y)\) otherwise. The function \(c_T (\nu, \eta)\) can be assumed to be zero if \(\nu (y) = \eta (y)\) for some \(y\). The process dynamics is determined by the infinitesimal transformation \[ \mathbb{P} [\eta_{t + dt} = \nu \text{ on } T \mid \eta_t] = c_T (\eta_t, \nu) dt + o(dt). \] It is assumed that \(c_T\) has a finite range: there exists an \(R < \infty\) so that \(c_T (\cdot, \cdot)\) is zero if \(T\) has length greater than \(R\) and such that for any \(x\) in \(\mathbb{Z}\) and \(T\) containing \(x\) of length at most \(R\), \(c_T (\nu, \eta)\) depends only on the values \(\eta (x - R)\), \(\eta (x - R + 1), \dots, \eta (x), \dots, \eta (x + R)\); and that function \(c_T\) has bounded flip rates: for each site \(x\), it is valid \(\sum_{y \in T} \sum_{\nu \in \mathbb{D}^T} c_T (\nu, \eta) < 1.\) The main results of the paper are the following theorems: 1. Let \(f (\eta_t) = S(t) f(\eta)\) be the transformation semigroup of the process. If \(t_n\) is a sequence of times tending to infinity such that \(f(\eta_{t_n}) \to v\) in distribution, then \(v\) is an invariant measure for the process. 2. If the process has a unique invariant measure, then it is ergodic.
scientific article; zbMATH DE number 784733