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

Albeverio S, Kondratiev Y, Röckner M (1997)
JOURNAL OF FUNCTIONAL ANALYSIS 149(2): 415-&.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
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.
Stichworte
quasi-invariant measures; Gibbs states; irreducibility of; extremality; Dirichlet forms; processes/diffusions; ergodicity of operator semigroups and of Markov; ergodicity of measures wrt shifts; classical and; quantum lattice models; Euclidean quantum field theory
Erscheinungsjahr
1997
Zeitschriftentitel
JOURNAL OF FUNCTIONAL ANALYSIS
Band
149
Ausgabe
2
Seite(n)
415-&
ISSN
0022-1236
Page URI
https://pub.uni-bielefeld.de/record/1627421

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Albeverio S, Kondratiev Y, Röckner M. Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. JOURNAL OF FUNCTIONAL ANALYSIS. 1997;149(2):415-&.
Albeverio, S., Kondratiev, Y., & Röckner, M. (1997). Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. JOURNAL OF FUNCTIONAL ANALYSIS, 149(2), 415-&.
Albeverio, S., Kondratiev, Y., and Röckner, M. (1997). Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. JOURNAL OF FUNCTIONAL ANALYSIS 149, 415-&.
Albeverio, S., Kondratiev, Y., & Röckner, M., 1997. Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. JOURNAL OF FUNCTIONAL ANALYSIS, 149(2), p 415-&.
S. Albeverio, Y. Kondratiev, and M. Röckner, “Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states”, JOURNAL OF FUNCTIONAL ANALYSIS, vol. 149, 1997, pp. 415-&.
Albeverio, S., Kondratiev, Y., Röckner, M.: Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. JOURNAL OF FUNCTIONAL ANALYSIS. 149, 415-& (1997).
Albeverio, Sergio, Kondratiev, Yuri, and Röckner, Michael. “Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states”. JOURNAL OF FUNCTIONAL ANALYSIS 149.2 (1997): 415-&.