We construct a family of probability spaces (f2, #", P γ \ γ > 0 associated with the Euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow φ t (time evolution) leaving P y globally invariant. φ t is obtained as the limit of Galerkin approximations associated with Euler equations. P γ is also in invariant measure for a stochastic process associated with a Navier-Stokes equation with viscosity y, stochastically perturbed by a white noise force. Contents 0. Introduction 431 1. The Euler Equation in Two Dimensions 433 1.1. General Setting 433 1.2. Invariant Quantities of the Motion 434 1.3. The Abstract Wiener Space Formulation 434 2. The Euler Flow and the Invariant Measures 437 2.1. The Invariant Measures 437 2.2. The Euler Flow 438 3. The Perturbed Navier-Stokes Equation 440 3.1. Invariant Measures 440 3.2. The Perturbed Navier-Stokes Flow

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The theory of integration in infinite dimensions is in some sense the backbone of probability theory. On this backbone the stochastic calculus of variations has given rise to the flesh of differential calculus. Its first step is the construction at each point of the probability space of a Cameron Martin-like tangent space in which the desired differential calculus can be developed. This construction proceeds along the lines of first-order differential geometry. In this paper we address the following questions: what could be the meaning of``curvature of the probability space'' how and why? How can curvatures be defined and computed? Why could a secondorder differential geometry be relevant to stochastic analysis? We try to answer these questions for the probability space associated to the Brownian motion of a compact Riemannian manifold. Why? A basic energy identity for anticipative stochastic integrals will be obtained as a byproduct of our computation of curvature. How? There are essentially four bottlenecks in the development of differential geometry on Wiener Riemann manifolds: (i) the difficulty of finding an atlas of local charts such that the changes of charts preserve the class of the Wiener-like measures together with their associated Cameron Martin-like tangent spaces; (ii) the difficulty of finding cylindrical approximations preserving the natural geometrical objects; (iii) the difficulty of renormalizing the divergent series to which the summation operations of finite dimensional differential geometry give rise in the non intrinsic context of local charts; (iv) the nonavailability of the computational procedures analogous to the local coordinates systems of the classical differential article no.

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We prove a Euler-Poincaré reduction theorem for stochastic processes taking values in a Lie group and we show examples of its application to SO(3) and to the group of diffeomorphisms.

We prove a variational principle for stochastic Lagrangian Navier-Stokes trajectories on manifolds. We study the behaviour of such trajectories concerning stability as well as rotation between particles; the two-dimensional torus case is described in detail.

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