Invariance implies Gibbsian: some new results
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.
scientific article; zbMATH DE number 2114604