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 Td, 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 on the group of volume preserving diffeomorphisms of T² a Riemannian structure which has a positive renormalized Ricci tensor.