Long-time contractivity estimates for kinetic Kolmogorov-Fokker-Planck equations

We prove long-time contractivity estimates and exponential rates of convergence to equilibrium for solutions of hypoelliptic diffusion equations, which include the well-known Kolmogorov equation and similar kinetic Fokker-Planck equations in ${\mathbb{R}}^d$. Compared to the existing literature, our proof exploits a different approach, elementary and self-contained, based on oscillation estimates for the adjoint problem. We first prove contractivity in Wasserstein distances through doubling variables (coupling) methods. Next, we upgrade the estimate to weighted $L^1$-(or total variation) norms, thanks to short-time hypocoercivity gradient estimates.