Invariance implies Gibbsian: some new results
type: Item
instance of: scholarly article
title: Invariance implies Gibbsian: some new results
zbMATH Open document ID: 1064.53021
author: Feng-Yu Wang
author: Vladimir I. Bogachev
author: Michael Röckner
published in: Communications in Mathematical Physics
publication date: 2004-11-11T00:00:00Z
review text: The authors investigate probability measures \(\mu\) on a connected Riemannian manifold \(M\) which satisfy \(L_Z^\ast\mu = 0\), where \(Z\) is a vector field on \(M\) and \(L_Zf = \triangle f + \langle Z, \nabla f\rangle\) for all \(f\in C^2(M)\). For example (theorem 2.4), assuming that \(M\) has Ricci curvature bounded from below, the balls of radii \(r\) (centred at all points of \(M\)) have volumes bounded from below by a positive constant, \(\| Z\| \in L^2(\mu)\), \(Z = \nabla U\) and \(U\in H^{1,1}_{\text{loc}}(\lambda _M)\), \(\lambda _M\) being the Riemannian volume on \(M\), they provide an explicit formula for \(\mu\): \(d\mu = \| \exp (U)\| ^{-1}_{L^1(\lambda _M)}\cdot\exp (U)\;d\lambda _{\mu}\). Similar problems on manifolds of infinite dimension are considered.
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 2114604
zbMATH Keywords: probability measure
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: laplacian
zbMATH Keywords: vector field
zbMATH Keywords: Ricci curvature
reviewed by: Paweł Walczak
MaRDI profile type: Publication
full work available at URL: s00220-004-1096-5
OpenAlex ID: W1971737366
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DOI: 10.1007/S00220-004-1096-5
description: scientific article; zbMATH DE number 2114604