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We formulate a stochastic least-action principle for solutions of theincompressible Navier-Stokes equation, which formally reduces to Hamilton'sprinciple for the incompressible Euler solutions in the case of zero viscosity.We use this principle to give a new derivation of a stochastic Kelvin Theoremfor the Navier-Stokes equation, recently established by Constantin and Iyer,which shows that this stochastic conservation law arises fromparticle-relabelling symmetry of the action. We discuss issues ofirreversibility, energy dissipation, and the inviscid limit of Navier-Stokessolutions in the framework of the stochastic variational principle. Inparticular, we discuss the connection of the stochastic Kelvin Theorem with ourprevious "martingale hypothesis" for fluid circulations in turbulent solutionsof the incompressible Euler equations.

Stochastic Least-Action Principle for the Incompressible Navier-Stokes Equation