Stochastic variational principles for dissipative equations with advected quantities

     We present symmetry reduction for material stochastic Lagrangian systems with advected quantities whose configuration space is a Lie group. Such variational principles yield deterministic as well as stochastic constrained variational principles for dissipative equations of motion in spatial representation. We apply this technique to the compressible Navier-Stokes equation and to magneto-hydrodynamics for charged viscous compressible fluids. A stochastic Kelvin-Noether theorem is derived. This talk is based on a joint work with Ana Bela Cruzeiro and Tudor Ratiu.