L2 geometric ergodicity for the kinetic Langevin process with non-equilibrium steady states

In non-equilibrium statistical physics models, the invariant measure $\mu$ of the process does not have an explicit density. In particular the adjoint $L^*$ in $L^2(\mu)$ of the generator $L$ is unknown and many classical techniques fail in this situation. An important progress has been made in [5] where functional inequalities are obtained for non-explicit steady states of kinetic equations under rather general conditions. However in [5] in the kinetic case the geometric ergodicity is only deduced from the functional inequalities for the case with conservative forces, corresponding to explicit steady states. In this note we obtain $L^2$ convergence rates in the non-equilibrium case.