Nonergodicity of Euler fluid dynamics on tori versus positivity of the Arnold-Ricci tensor

Brownian motions above the group G of volume preserving diffeomorphisms of the torus T-d, d >= 2, are constructed. The asymptotic behaviour for large time of those processes shows the nonexistence of a probability measure invariant under the deterministic incompressible fluid dynamics. The energy induces won the group of volume preserving diffeomorphisms of T-2 a Riemannian structure which has a positive renormalized Ricci tensor. (c) 2007 Elsevier Inc. All rights reserved.