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

We prove long-time contractivity estimates and exponential rate of convergence to equilibrium for solutions of hypoelliptic diffusion equations, which include the well-known Kolmogorov equation and similar kinetic Fokker–Planck equations in Rd. 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-variable (coupling) methods. Next, we upgrade the estimate to weighted L1-(or total variation) norms, thanks to short-time hypocoercivity gradient estimates.