Hypocoercivity meets lifts

TL;DR

This paper unifies hypocoercivity and second-order lifts of diffusion processes, providing a constructive framework applicable to kinetic equations, and analyzes the convergence rates of Langevin-type dynamics, including GLE.

Contribution

It introduces a unified, constructive hypocoercivity framework for second-order lifts, applicable to a broad class of linear kinetic equations, with detailed analysis of Langevin dynamics.

Findings

Adaptive Langevin dynamics is near-optimal for quadratic potentials.

The Generalised Langevin Equation is a second-order lift with ballistic convergence limits.

Explicit Gaussian case computations illustrate convergence rate phenomena.

Abstract

We unify the variational hypocoercivity framework established by D. Albritton, S. Armstrong, J.-C. Mourrat, and M. Novack, with the notion of second-order lifts of reversible diffusion processes, recently introduced by A. Eberle and F. L\"orler. We give an abstract, yet fully constructive, presentation of the theory, so that it can be applied to a large class of linear kinetic equations. As this hypocoercivity technique does not twist the reference norm, we can recover accurate and sharp convergence rates in various models. Among those, adaptive Langevin dynamics (ALD) is discussed in full detail and we show that for near-quadratic potentials, with suitable choices of parameters, it is a near-optimal second-order lift of the overdamped Langevin dynamics. As a further consequence, we observe that the Generalised Langevin Equation (GLE) is a also a second-order lift, as the standard (kinetic) Langevin dynamics are, of the overdamped Langevin dynamics. Then, convergence of (GLE) cannot exceed ballistic speed, i.e. the square root of the rate of the overdamped regime. We illustrate this phenomenon with explicit computations in a benchmark Gaussian case.