## Electronic Journal of Probability

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

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

#### 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**

Received: 6 April 2017

Accepted: 2 October 2017

First available in Project Euclid: 20 October 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/17-EJP114

**Mathematical Reviews number (MathSciNet)**

MR3718719

**Zentralblatt MATH identifier**

06797901

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65]

Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60H30: Applications of stochastic analysis (to PDE, etc.) 35K65: Degenerate parabolic equations 60F05: Central limit and other weak theorems 37A25: Ergodicity, mixing, rates of mixing

**Keywords**

one-dimensional diffusions hypoellipticity duality by intertwining Bessel 3 process Hörmander’s density theorem strong stationary times

**Rights**

Creative Commons Attribution 4.0 International License.

#### 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