  Smooth solutions of the incompressible Euler equations are characterized by
the property that circulation around material loops is conserved. This is the
Kelvin theorem. Likewise, smooth solutions of Navier-Stokes are characterized
by a generalized Kelvin's theorem, introduced by Constantin-Iyer (2008). In
this note, we introduce a class of stochastic fluid equations, whose smooth
solutions are characterized by natural extensions of the Kelvin theorems of
their deterministic counterparts, which hold along certain noisy flows. These
equations are called the stochastic Euler-Poincar\'{e} and stochastic
Navier-Stokes-Poincar\'{e} equations respectively. The stochastic
Euler-Poincar\'{e} equations were previously derived from a stochastic
variational principle by Holm (2015), which we briefly review. Solutions of
these equations do not obey pathwise energy conservation/dissipation in
general. In contrast, we also discuss a class of stochastic fluid models,
solutions of which possess energy theorems but do not, in general, preserve
circulation theorems.
