Construction of Markov processes of certain spin systems on \(Z\) (Q1816454)

Jump-type Markov processes of \({\mathbf R}\)-valued spin fields on \({\mathbf Z}\) are constructed. In a process an \({\mathbf R}\)-valued spin is attached to each site of \({\mathbf Z}\) and the values of spins evolve at random times. The values of spins on each pair of adjacent sites evolve simultaneously according to a certain law and the jump rate is an unbounded function of the values of spins. The processes in equilibrium and in non-equilibrium are constructed, separately, under suitable conditions for an initial distribution. It is also shown that the process has a finite cluster property if the initial distribution, in addition to the assumption for the construction, is ergodic relative to the shift transformation on the configuration space.