To visualize how the randomness of a Markov process $X$ is spreading, one can consider subset-valued dual processes $I$ constructed by intertwining. In the framework of one-dimensional diffusions, we investigate the behavior of such dual processes $I$ in the presence of hypoellipticity for $X$. The Pitman type property asserting that the measure of $I$ is a time-changed Bessel 3 process is preserved, the effect of hypoellipticity is only found at the level of the time change. It enables to recover the density theorem of Hörmander in this simple degenerate setting, as well as to construct strong stationary times by introducing different dual processes.
"Duality and hypoellipticity: one-dimensional case studies." Electron. J. Probab. 22 1 - 32, 2017. https://doi.org/10.1214/17-EJP114