article
Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states
published
yes
Sergio
Albeverio
author
Yuri
Kondratiev
author 15419
Michael
Röckner
author 10585
10020
department
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.t<is an) element of>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 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.
ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS1997
eng
quasi-invariant measuresGibbs statesirreducibility ofextremalityDirichlet formsprocesses/diffusionsergodicity of operator semigroups and of Markovergodicity of measures wrt shiftsclassical andquantum lattice modelsEuclidean quantum field theory
JOURNAL OF FUNCTIONAL ANALYSIS
0022-1236
A1997XZ2210000510.1006/jfan.1997.3099
1492415-&
Albeverio S, Kondratiev Y, Röckner M (1997) <br />Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states.<br />JOURNAL OF FUNCTIONAL ANALYSIS 149(2): 415-&.
Albeverio S, Kondratiev Y, Röckner M. Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. <em>JOURNAL OF FUNCTIONAL ANALYSIS</em>. 1997;149(2):415-&.
<div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Albeverio, S., Kondratiev, Y., & Röckner, M. (1997). Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. <em>JOURNAL OF FUNCTIONAL ANALYSIS</em>, <em>149</em>(2), 415-&.</div>
<div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Albeverio, Sergio, Kondratiev, Yuri, and Röckner, Michael. 1997. “Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states”. <em>JOURNAL OF FUNCTIONAL ANALYSIS</em> 149 (2): 415-&.</div>
S. Albeverio, Y. Kondratiev, and M. Röckner, “Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states”, <em>JOURNAL OF FUNCTIONAL ANALYSIS</em>, vol. 149, 1997, pp. 415-&.
Albeverio S, Kondratiev Y, Röckner M (1997) <br /><em>JOURNAL OF FUNCTIONAL ANALYSIS</em> 149(2): 415-&.
Albeverio, Sergio, Kondratiev, Yuri, and Röckner, Michael. “Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states”. <em>JOURNAL OF FUNCTIONAL ANALYSIS</em> 149.2 (1997): 415-&.
<div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Albeverio, S., Kondratiev, Y. & Röckner, M. (1997). Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. <em>JOURNAL OF FUNCTIONAL ANALYSIS</em>, <em>149</em>(2), 415-&. ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS.</div>
Albeverio, S., Kondratiev, Y., & Röckner, M. (1997). Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. <em>JOURNAL OF FUNCTIONAL ANALYSIS</em>, <em>149</em>(2), 415-&.
S. Albeverio, Y. Kondratiev, and M. Röckner, “Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states”, <em>JOURNAL OF FUNCTIONAL ANALYSIS</em>, <strong>1997</strong>, <em>149</em>, 415-&.
Albeverio, S.; Kondratiev, Y.; Röckner, M. (1997): Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states <em>JOURNAL OF FUNCTIONAL ANALYSIS</em>,149:(2): 415-&.
Albeverio, S., Kondratiev, Y., & Röckner, M., 1997. Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. <em>JOURNAL OF FUNCTIONAL ANALYSIS</em>, 149(2), p 415-&.
Albeverio, S., Kondratiev, Y., and Röckner, M. (1997). Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. <em>JOURNAL OF FUNCTIONAL ANALYSIS</em> 149, 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).
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