We develop the Clebsch variational method of deriving stochastic differential equations whose solutions follow the flow of a stochastic nonlinear Lie algebra operation on a configuration manifold. In the stochastic Clebsch action principle presented here, the noise couples to the phase space variables through a momentum map. This special coupling simplifies the structure of the resulting stochastic Hamilton equations. In particular, these stochastic Hamilton equations collectivize for Hamiltonians which depend only on the momentum map variable. The Stratonovich equations are derived from the Clebsch variational principle and then converted into It\^o form. We compare the Stratonovich and It\^o forms of the equations governing the components of the momentum map. 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.
