  We formulate a stochastic least-action principle for solutions of the
incompressible Navier-Stokes equation, which formally reduces to Hamilton's
principle for the incompressible Euler solutions in the case of zero viscosity.
We use this principle to give a new derivation of a stochastic Kelvin Theorem
for the Navier-Stokes equation, recently established by Constantin and Iyer,
which shows that this stochastic conservation law arises from
particle-relabelling symmetry of the action. We discuss issues of
irreversibility, energy dissipation, and the inviscid limit of Navier-Stokes
solutions in the framework of the stochastic variational principle. In
particular, we discuss the connection of the stochastic Kelvin Theorem with our
previous "martingale hypothesis" for fluid circulations in turbulent solutions
of the incompressible Euler equations.
