The ergodic properties of two stochastic models I and II are investigated. Each model is described by a fieldx(t),t > 0. The probability measures on X 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.