  In this paper it is proved that for families of stochastic operators on a
countable tensor product, depending smoothly on parameters, any spectral
projection persists smoothly, where smoothness is defined using norms based on
ideas of Dobrushin. A rigorous perturbation theory for families of stochastic
operators with spectral gap is thereby created. It is illustrated by deriving
an effective slow 2-state dynamics for a 3-state probabilistic cellular
automaton. Some further potential applications are discussed.
