  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;
moreover, the most likely value for velocity is not zero.
