Hypocoercivity meets lifts

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. Lörler. 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.

17 pages

References in corpus (10)

• Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps
• The Zig-Zag Process and Super-Efficient Sampling for Bayesian Analysis of Big Data
• Asymptotic analysis for the generalized langevin equation
• Ergodicity of the zigzag process
• On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics
• Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes
• Variational methods for the kinetic Fokker-Planck equation
• On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms
• Scaling limits for the generalized Langevin equation
• Non-reversible lifts of reversible diffusion processes and relaxation times