TY - JOUR
AB - 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.tR) 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.
AU - Albeverio, Sergio
AU - Kondratiev, Yuri
AU - Röckner, Michael
ID - 1627421
IS - 2
JF - JOURNAL OF FUNCTIONAL ANALYSIS
KW - quasi-invariant measures
KW - Gibbs states
KW - irreducibility of
KW - extremality
KW - Dirichlet forms
KW - processes/diffusions
KW - ergodicity of operator semigroups and of Markov
KW - ergodicity of measures wrt shifts
KW - classical and
KW - quantum lattice models
KW - Euclidean quantum field theory
SN - 0022-1236
TI - Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states
VL - 149
ER -