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Hypocoercivity meets lifts

Giovanni Brigati, Francis Lörler, Lihan Wang

TL;DR

The paper develops a unified variational hypocoercivity framework that ties together non-reversible lifts with classical hypocoercivity for linear kinetic equations of the form $\partial_t f + \mathcal{T} f = \mathrm{L}_v f$ under a (potentially non-separable) invariant measure. It provides an explicit exponential convergence bound with constructive rate $\lambda$ and constant $C$ using a time-averaged $L^2$ energy and a space-time-velocity Poincaré-type inequality, while clarifying the role of the lift structure in obtaining both upper and lower bounds on the rate. The framework is illustrated through second-order lifts of overdamped Langevin dynamics, notably adaptive Langevin dynamics (ALD) and the generalized Langevin equation (GLE); results show near-quadratic potentials yield near-optimal rates for ALD, and establish fundamental limits on rate acceleration achievable by lifts via lower bounds in the GLE case. Overall, the work links variational hypocoercivity with non-reversible lifting theory to produce sharp, constructible convergence rates for a broad class of kinetic models, with implications for designing accelerated sampling dynamics.

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.

Hypocoercivity meets lifts

TL;DR

The paper develops a unified variational hypocoercivity framework that ties together non-reversible lifts with classical hypocoercivity for linear kinetic equations of the form under a (potentially non-separable) invariant measure. It provides an explicit exponential convergence bound with constructive rate and constant using a time-averaged energy and a space-time-velocity Poincaré-type inequality, while clarifying the role of the lift structure in obtaining both upper and lower bounds on the rate. The framework is illustrated through second-order lifts of overdamped Langevin dynamics, notably adaptive Langevin dynamics (ALD) and the generalized Langevin equation (GLE); results show near-quadratic potentials yield near-optimal rates for ALD, and establish fundamental limits on rate acceleration achievable by lifts via lower bounds in the GLE case. Overall, the work links variational hypocoercivity with non-reversible lifting theory to produce sharp, constructible convergence rates for a broad class of kinetic models, with implications for designing accelerated sampling dynamics.

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.

Paper Structure

This paper contains 9 sections, 5 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Setting and main goal
  3. From hypocoercivity to non-reversible lifts
  4. Assumptions and discussion
  5. Main result
  6. Examples
  7. The general convergence proof
  8. Example: Hypocoercivity of adaptive Langevin dynamics
  9. Lower bounds for the generalised Langevin equation

Key Result

Theorem 1

Let $f(t,x,v)$ be the solution of eq:genkineq with initial condition $f(t=0,x,v) = f_0(x,v) \in L^2(\hat{\mu})$. Then as $t\to \infty$, $f_t$ converges exponentially fast to $(f_0):= \int f_0(x,v) \mathop{}\!\mathrm{d} \hat{\mu}(x,v)$. More precisely, there exist constants $C>1,\lambda>0$ such that In particular, suppose there exists some $T_*>0$ such that then we have $C=\exp(T_*\lambda_{T_*})$

Theorems & Definitions (15)

  • Definition 1: eberle2024non
  • Remark 1.1: Lower bounds for lifts
  • Remark 1.2: DMS Hypocoercivity and lifts
  • Remark 1.3: Variational hypocoercivity and DMS
  • Theorem 1
  • Remark 1.4
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm:expconv']}
  • Proposition 3.1
  • ...and 5 more