We consider the kinetic Brownian motion in the Euclidean plane together with its velocity in the circle. In this talk I will present a Bismut-type formula for the semi-group of this hypoelliptic process. The result is based on the expansion of the Brownian motion and the explicit computation of Malliavin dual in Gaussian space.

I will also give some applications: gradient inequalities in large times for the semi-group as well as Liouville property for the generator of the kinetic Brownian motion.

This is a joint work with Marc Arnaudon, Michel Bonnefont and Delphine Féral (IMB, Bordeaux).