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Ergodicity for the stochastic dynamics of quasiinvariant measures with applications to Gibbs states
, 1997
"... The convex set M a of quasiinvariant measures on a locally convex space E with given "shift"RadonNikodym derivatives (i.e., cocycles) a = (a tk ) k2K 0 ; t2R is analyzed. The extreme points of M a are characterized and proved to be nonempty. A specification (of lattice type) is co ..."
Abstract

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The convex set M a of quasiinvariant measures on a locally convex space E with given "shift"RadonNikodym derivatives (i.e., cocycles) a = (a tk ) k2K 0 ; t2R is analyzed. The extreme points of M a are characterized and proved to be nonempty. 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 wellknown method due to DynkinFollmer a unique representation of an arbitrary element in M a in terms of extreme ones is derived. Furthermore, the corresponding classical Dirichlet forms (E ; D(E )) and their associated semigroups (T t ) t?0 on L 2 (E; ) are discussed. Under a mild positivity condition it is shown that 2 M a is extreme if and only if (E ; D(E )) is irreducible or equivalently, (T t ) t?0 is ergodic. This implies timeergodicity of associated diffusions. Applications to Gibbs states of classical and quantum lattice models as well as those occuring in Euclidean...
Ergodicity of L²Semigroups and Extremality of Gibbs States
, 1995
"... We extend classical results of HolleyStroock on the characterization of extreme Gibbs states for the Ising model in terms of the irreducibility (resp. ergodicity) of the corresponding Glauber dynamics to the case of lattice systems with unbounded (linear) spin spaces. We first develop a general fra ..."
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We extend classical results of HolleyStroock on the characterization of extreme Gibbs states for the Ising model in terms of the irreducibility (resp. ergodicity) of the corresponding Glauber dynamics to the case of lattice systems with unbounded (linear) spin spaces. We first develop a general framework to discuss questions of this type using classical Dirichlet forms on infinite dimensional state spaces and their associated diffusions. We then describe concrete applications to lattice models with polynomial interactions (i.e., the discrete P(&phi;)dmodels of Euclidean quantum field theory). In addition, we prove the equivalence of extremality and shiftergodicity for tempered Gibbs states of these models and also discuss this question in the general framework.