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

TL;DR

This paper establishes that for certain diffusions on , contractivity of the drift guarantees exponential convergence in Wasserstein distances, with applications to Langevin diffusions and oscillator chains.

Contribution

It proves the equivalence between drift contractivity and Wasserstein contraction for non-reversible diffusions, extending known results to generalized Langevin diffusions.

Findings

Wasserstein distances contract at a constant rate under drift contractivity.

Concentration inequalities are derived for ergodic averages.

Results apply to non-equilibrium oscillator chains and generalized Langevin diffusions.

Abstract

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.