  We derive stochastic differential equations whose solutions follow the flow
of a stochastic nonlinear Lie algebra operation on a configuration manifold.
For this purpose, we develop a stochastic Clebsch action principle, in which
the noise couples to the phase space variables through a momentum map. This
special coupling simplifies the structure of the resulting stochastic Hamilton
equations for the momentum map. In particular, these stochastic Hamilton
equations collectivize for Hamiltonians that depend only on the momentum map
variable. The Stratonovich equations are derived from the Clebsch variational
principle and then converted into It\^o form. In comparing the Stratonovich and
It\^o forms of the stochastic dynamical equations governing the components of
the momentum map, we find that the It\^o contraction term turns out to be a
double Poisson bracket. Finally, we present the stochastic Hamiltonian
formulation of the collectivized momentum map dynamics and derive the
corresponding Kolmogorov forward and backward equations.
