Ergodicity for the Stochastic Dynamics of Quasi-invariant Measures with Applications to Gibbs States
✍ Scribed by Sergio Albeverio; Yuri G. Kondratiev; Michael Röckner
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 621 KB
- Volume
- 149
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
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 # K 0 , t # 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 Fo llmer 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 & # 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 quantum field theory are presented. In particular, it is proved that the stochastic quantization of a Guerra Rosen Simon Gibbs state on D$(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.