The convergence of the biased annihilating branching process and the double-flipping process in Zd

It is shown that, if the initial measure is translation-invariant, then finite-range stochastic Ising models allowing zero flip-rates converge. In particular, the biased annihilating process converges to a mixture of a product measure and δø and the double-flipping process converges to a product measure. thod of relative entropy is employed.