  In view of studying incompressible inviscid fluids, Brenier introduced in the
late 80's a relaxation of a geodesic problem addressed by Arnold in 1966.
Instead of inviscid fluids, the present paper is devoted to incompressible
viscid fluids. A natural analogue of Brenier's problem is introduced, where
generalized flows are no more supported by absolutely continuous paths, but by
Brownian sample paths. It turns out that this new variational problem is an
entropy minimization problem with marginal constraints entering the class of
convex minimization problems. This paper explores the connection between this
variational problem and Brenier's original problem. Its dual problem is derived
and the general shape of its solution is described. Under the restrictive
assumption that the pressure is a nice function, the kinematics of its solution
is made explicit and its connection with the Navier-Stokes equation is
established.
