Abstract
The ergodic properties of two stochastic models ∑I and ∑II are investigated. Each model is described by a fieldx(t),t ⩾> 0, on the lattice г =Zd,d < ∞. For ∑I,x(t) evolves according to the equations wherexs(t) eR for eachs eF. Here the {ws(t): s e Г} are independent, one-dimensional Wiener processes, Φ2 is a bounded interaction between adjacent lattice sites, and the potentials Φ1 and Φ2 satisfy appropriate regularity conditions. It is shown that for each model,x(t) is a Markov process on an infinite-dimensional phase spaceX. The probability measures onX that satisfy the Dobrushin-Lanford-Ruelle (DLR) conditions are stationary for this process and have a mixing property. Moreover, for ∑I any stationary, time-reversal-invariant probability measure that has certain regularity properties must satisfy the DLR conditions.
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