The Langevin equation on Lie algebras: Maxwell–Boltzmann is not always the equilibrium

We give a geometric formulation of the Fokker–Planck–Kramer equations for a particle moving on a Lie algebra under the influence of a dissipative and a random force. Special cases of interest are fluid mechanics, the Stochastic Loewner equation and the rigid body. We find that the Boltzmann distribution, although a static solution, is not normalizable when the algebra is not unimodular. This is because the invariant measure of integration in momentum space is not the standard one. We solve the special case of the upper half-plane (hyperboloid) explicitly: there is another equilibrium solution to the Fokker–Planck equation, which is integrable. It breaks rotation invariance.