The Posterior metric and the Goodness of Gibbsianness for transforms of Gibbs measures

Abstract

We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models subjected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations. Assuming no a priori metric on the local state spaces but only a measurable structure, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by Dobrushin uniqueness here). We show in the concrete example of the time evolution of rotators on the $(q-1)$-dimensional sphere how this method can be used to obtain estimates in terms of the familiar Euclidean metric. In another application we prove the preservation of Gibbsianness for sufficiently fine local coarse-grainings when the Hamiltonian satisfies a Lipschitz property