  Let $Q$ be a Riemannian $G$-manifold. This paper is concerned with the
symmetry reduction of Brownian motion in $Q$ and ramifications thereof in a
Hamiltonian context. Specializing to the case of polar actions we discuss
various versions of the stochastic Hamilton-Jacobi equation associated to the
symmetry reduction of Brownian motion and observe some similarities to the
Schr\"odinger equation of the quantum free particle reduction as described by
Feher and Pusztai. As an application we use this reduction scheme to derive
examples of quantum Calogero-Moser systems from a stochastic setting.
