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

Understand

In the case of diffusions on $\mathbb R^d$ with constant diffusion matrix, without assuming reversibility nor hypoellipticity, we prove that the contractivity of the deterministic drift is equivalent to the constant rate contraction of Wasserstein distances $\mathcal W_p$, $p\in[1,\infty]$.

• It also implies concentration inequalities for ergodic means of the process.
• Such a contractivity property is then established for some non-equilibrium chains of anharmonic oscillators and for some generalised Langevin diffusions when the potential is convex with bounded Hessian and the friction is sufficiently high.
• This extends previous known results for the usual (kinetic) Langevin diffusion.