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In the case of diffusions on $\mathbb R^d$ with constant diffusion matrix,without assuming reversibility nor hypoellipticity, we prove that thecontractivity of the deterministic drift is equivalent to the constant ratecontraction of Wasserstein distances $\mathcal W_p$, $p\in[1,\infty]$. It alsoimplies concentration inequalities for ergodic means of the process. Such acontractivity property is then established for some non-equilibrium chains ofanharmonic oscillators and for some generalised Langevin diffusions when thepotential is convex with bounded Hessian and the friction is sufficiently high.This extends previous known results for the usual (kinetic) Langevin diffusion.

Almost sure contraction for diffusions on $\mathbb R^d$. Application to generalised Langevin diffusions