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Invariance Implies Gibbsian: Some New Results

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Abstract

We investigate stationary distributions of stochastic gradient systems in Riemannian manifolds and prove that, under certain assumptions, such distributions are symmetric. These results are extended to countable products of finite dimensional manifolds and applied to Gibbs distributions in the case where the single spin spaces are Riemannian manifolds. In particular, we obtain a new result concerning the question whether all invariant measures are Gibbsian. Actually, we consider a more general object: weak elliptic equations for measures, which, on the one hand, yields the results obtained stronger than the above mentioned statements, and, on the other hand, enables us to give simpler proofs of more general than previously known facts. Applications to concrete models of lattice systems over ℤ d with not necessarily compact spin space are presented (also in the case d ≥ 3 under certain assumptions of decay of interaction).

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, 119899, Moscow, Russia

    Vladimir I Bogachev

  2. Fakultät für Mathematik, Universität Bielefeld, 33615, Bielefeld, Germany

    Michael Röckner

  3. Department of Mathematics, Beijing Normal University, Beijing, 100875, P.R. China

    Feng-Yu Wang

Authors
  1. Vladimir I Bogachev
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  2. Michael Röckner
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  3. Feng-Yu Wang
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Communicated by J.L. Lebowitz

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Bogachev, V., Röckner, M. & Wang, FY. Invariance Implies Gibbsian: Some New Results. Commun. Math. Phys. 248, 335–355 (2004). https://doi.org/10.1007/s00220-004-1096-5

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  • DOI: https://doi.org/10.1007/s00220-004-1096-5

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