<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup></mml:math> hypocoercivity, deviation bounds, hitting times and Lyapunov functions

We establish that, for a Markov semi-group, $L^2$ hypocoercivity, i.e. contractivity modified norm, implies quantitative deviation bounds additive functionals of the associated process and exponential integrability hitting time sets with positive measure. Moreover, in case diffusion processes under strong hypoellipticity assumption, we prove that hypocoercivity existence Lyapunov function generator.