A conditioned hypoelliptic process on a compact manifold, satisfying the strong Hörmander’s condition, is a hypoelliptic bridge. If the Markov generator satisfies the two step strong Hörmander condition, the drift of the conditioned hypoelliptic bridge is integrable on $[0,1]$ and the hypoelliptic bridge is a continuous semi-martingale.
"On hypoelliptic bridge." Electron. Commun. Probab. 21 1 - 12, 2016. https://doi.org/10.1214/16-ECP4646