×

Conditions for the nonergodicity of one-dimensional Stavskaya media. (English. Russian original) Zbl 0573.60061

Sel. Math. Sov. 4, 63-71 (1985); translation from Interacting Markov processes and their applications to biology, Pushchino 1979, 37-48 (1979).
Consider a homogeneous Markov chain, with states \(x=(x_ h)\), which are mappings from \({\mathbb{Z}}\) to the finite set \(N=\{0,1,...,n\}\). It is determined by a (transition) operator P, assumed homogeneous, local, and with independent transitions. An operator P is called determinate if it maps every point measure to another point measure. Operators for which the measure \(\bar n \)(the measure concentrated on the state ”\(x_ h=n\) for all h”) is invariant are called Stavskaya operators.
In this paper, the author considers Stavskaya operators P which are represented as \(P=SQ\), with Q determinate and S a noise operator (defined by: if \(x_ h=n\), then \((S_ x)_ h(n)=1\) and if \(x_ h\neq i\), then \((S_ x)_ h(i)<1)\). Such operators clearly have the property \(P\bar n=\bar n\), and the present paper provides conditions under which P is nonergodic, i.e. possesses invariant measures, different from \(\bar n.\) In particular, it does not assume that \(Q\bar O=\bar O\), or that Q be monotonic.
Reviewer: W.Wreszinski

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60B99 Probability theory on algebraic and topological structures
PDFBibTeX XMLCite