Ergodicity for the Stochastic Dynamics of Quasi-invariant Measures with Applications to Gibbs States

Abstract

The convex set Maof quasi-invariant measures on a locally convex spaceEwith given “shift”-Radon–Nikodym derivatives (i.e., cocycles)a=(atk)k∈K0,t∈Ris analyzed. The extreme points of Maare characterized and proved to be non-empty. A specification (of lattice type) is constructed so that Macoincides with the set of the corresponding Gibbs states. As a consequence, via a well known method due to Dynkin and Föllmer a unique representation of an arbitrary element in Main terms of extreme ones is derived. Furthermore, the corresponding classical Dirichlet forms (Eν,D(Eν)) and their associated semigroups (Tνt)t>0onL2(E;ν) are discussed. Under a mild positivity condition it is shown thatν∈Mais extreme if and only if (Eν,D(Eν)) is irreducible or equivalently, (Tνt)t>0is 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′(R2) ininfinite volumewith polynomial interaction is ergodic if the Gibbs state is extreme (i.e., is a pure phase), which solves a long-standing open problem.