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How to construct decay rates for kinetic Fokker--Planck equations?

Giovanni Brigati, Gabriel Stoltz

TL;DR

The paper analyzes decay to equilibrium for Vlasov–Fokker–Planck equations with a general separable Hamiltonian $H(x,v)=\phi(x)+\psi(v)$. By combining time-averaged $L^2$ entropies with a space-time Lions–Poincaré inequality and a constructive averaging lemma, it obtains fully explicit exponential and algebraic decay rates as well as dimension- and parameter-dependent constants. A key novelty is the explicit, constructive treatment of the divergence equation underlying the Lions inequality, enabling precise constants and scalable estimates even for non-quadratic kinetic energies and non-Gaussian momenta tails. The results provide a versatile framework for obtaining quantitative hypocoercivity estimates with direct implications for sampling and molecular dynamics, including how rates scale with friction, time averaging, and dimension. Overall, the work offers a practical, fully explicit methodology to bound time-averaged dissipation and decay in kinetic Fokker–Planck type models.

Abstract

We study time averages for the norm of solutions to kinetic Fokker--Planck equations associated with general Hamiltonians. We provide fully explicit and constructive decay estimates for systems subject to a confining potential, allowing fat-tail, sub-exponential and (super-)exponential local equilibria, which also include the classic Maxwellian case. The key step in our estimates is a modified Poincaré inequality, obtained via a Lions--Poincaré inequality and an averaging lemma.

How to construct decay rates for kinetic Fokker--Planck equations?

TL;DR

The paper analyzes decay to equilibrium for Vlasov–Fokker–Planck equations with a general separable Hamiltonian . By combining time-averaged entropies with a space-time Lions–Poincaré inequality and a constructive averaging lemma, it obtains fully explicit exponential and algebraic decay rates as well as dimension- and parameter-dependent constants. A key novelty is the explicit, constructive treatment of the divergence equation underlying the Lions inequality, enabling precise constants and scalable estimates even for non-quadratic kinetic energies and non-Gaussian momenta tails. The results provide a versatile framework for obtaining quantitative hypocoercivity estimates with direct implications for sampling and molecular dynamics, including how rates scale with friction, time averaging, and dimension. Overall, the work offers a practical, fully explicit methodology to bound time-averaged dissipation and decay in kinetic Fokker–Planck type models.

Abstract

We study time averages for the norm of solutions to kinetic Fokker--Planck equations associated with general Hamiltonians. We provide fully explicit and constructive decay estimates for systems subject to a confining potential, allowing fat-tail, sub-exponential and (super-)exponential local equilibria, which also include the classic Maxwellian case. The key step in our estimates is a modified Poincaré inequality, obtained via a Lions--Poincaré inequality and an averaging lemma.
Paper Structure (24 sections, 15 theorems, 182 equations)

This paper contains 24 sections, 15 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structural assumptions
  4. An averaging lemma
  5. Exponential convergence rates
  6. Algebraic convergence rates
  7. Weighted Lions' inequalities
  8. Conclusions
  9. Scaling with respect to the dimension
  10. Scaling with respect to the time average parameter
  11. Scaling with respect to the friction
  12. Notation and preliminary technical results
  13. Modified Poincaré inequalities and exponential decay rates
  14. Proof of Lemma \ref{['avglem']}
  15. Proof of Theorem \ref{['thm1new']}
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\psi(v) = \frac{|v|^2}{2}$ and let $\phi$ be a smooth spatial potential, which grows linearly or faster as $|x| \to \infty$. Then, for all $\tau>0$, there exists an explicit constant $\lambda(\tau)>0$ such that all solutions to VOU satisfy

Theorems & Definitions (29)

  • Theorem 1: Informal statement
  • Remark 2
  • Lemma 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Corollary 7
  • Remark 8
  • Remark 9
  • Theorem 10
  • ...and 19 more