For the Heisenberg group Brownian motion, the ratio of tube probabilities around horizontal paths phi and psi converges to exp(-1/2||phi||^2 + 1/2||psi||^2), giving Onsager-Machlup functional L(p,v) = -1/2||v||^2.
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On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group
For the Heisenberg group Brownian motion, the ratio of tube probabilities around horizontal paths phi and psi converges to exp(-1/2||phi||^2 + 1/2||psi||^2), giving Onsager-Machlup functional L(p,v) = -1/2||v||^2.