Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states

The convex set M-a of quasi-invariant measures on a locally convex space E with given ''shift''-Radon-Nikodym derivatives (i.e., cocycles) a = (a(tk))(k is an element of K0.t<is an) element of>R) is analyzed. The extreme points of M-a are characterized and proved to be non-empty. A specification (of lattice type) is constructed so that M-a coincides with the set of the corresponding Gibbs states. As a consequence, via a well known method due to Dynkin and Follmer a unique representation of an arbitrary element in M-a in terms of extreme ones is derived. Furthermore, the corresponding classical Dirichlet forms (E-v, D(E-v)) and their associated semigroups (T-t(v))(t>0) on L-2(E; v) are discussed. Under a mild positivity condition it is shown that v is an element of M-a is extreme if and only if (E-v, D(E-v)) is irreducible or equivalently, (T-t(v))(t>0) is ergodic. This implies time-ergodicity of associated diffusions. Applications to Gibbs states of classical and quantum lattice models as well as those occuring in Euclidean quantum field theory are presented. In particular, it is proved that the stochastic quantization of a Guerra-Rosen-Simon Gibbs state on D-1(R-2) in infinite volume with polynomial interaction is ergodic if the Gibbs state is extreme (i.e., is a pure phase), which solves a long-standing open problem. (C) 1997 Academic Press.