  In this paper, we extend the energy-Casimir stability method for
deterministic Lie-Poisson Hamiltonian systems to provide sufficient conditions
for the stability in probability of stochastic dynamical systems with
symmetries and multiplicative noise. We illustrate this theory with classical
examples of coadjoint motion, including the rigid body, the heavy top and the
compressible Euler equation in two dimensions. The main result of this
extension is that stable deterministic equilibria remain stable in probability
up to a certain stopping time which depends on the amplitude of the noise for
finite dimensional systems and on the amplitude the spatial derivative of the
noise for infinite dimensional systems.
