Stochastic Implicit Lagrange-Poincar\'e Reduction

In this paper we consider reduction of the stochastic Hamilton-Pontryagin principle formulated on the Pontryagin bundle of a manifold $Q$. We prove that a stochastic action invariant under the free and proper action of a Lie group $G$ drops to a reduced variational principle expressed in terms of variables of the Pontryagin bundle of the reduced space $Q/G$, the associated adjoint bundle $\tilde{\mathfrak{g}}:= (Q\times \mathfrak{g})/G$ and its dual bundle $\tilde{\mathfrak{g}}^*$. This provides a stochastic analogue of the deterministic implicit Lagrange-Poincar\'e reduction. The stochastic Euler-Lagrange equations drop to a set of stochastic horizontal and vertical Lagrange-Poincar\'e equations on $T(Q/G)\oplus T^*(Q/G)\oplus\tilde{\mathfrak{g}}\oplus\tilde{\mathfrak{g}}^*$. As examples, we consider stochastic perturbations of the rigid body with a rotor, as well as a Kaluza-Klein description of stochastic perturbations of a charged particle in a magnetic field.