## Electronic Journal of Probability

### Duality and hypoellipticity: one-dimensional case studies

Laurent Miclo

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 91, 32 pp.

Dates
Accepted: 2 October 2017
First available in Project Euclid: 20 October 2017

https://projecteuclid.org/euclid.ejp/1508464837

Digital Object Identifier
doi:10.1214/17-EJP114

Mathematical Reviews number (MathSciNet)
MR3718719

Zentralblatt MATH identifier
06797901

#### Citation

Miclo, Laurent. Duality and hypoellipticity: one-dimensional case studies. Electron. J. Probab. 22 (2017), paper no. 91, 32 pp. doi:10.1214/17-EJP114. https://projecteuclid.org/euclid.ejp/1508464837

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