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Convergence to a non-explicit steady state in non-factorized kinetic Fokker-Planck equations with (very) weak velocity confinements

Emeric Bouin, Luca Ziviani

TL;DR

The paper proves existence and uniqueness of a non-explicit steady state for a kinetic Fokker-Planck equation with strong space confinement and fat-tailed local equilibria, and establishes convergence toward this state in weighted $\mathsf{L}^1$ spaces. The authors develop weak Lyapunov functionals tailored to sub-Gaussian and polynomial tails and verify a positivity condition via kinetic Harnack inequalities, enabling a sub-geometric Harris-type convergence analysis. Depending on tail behavior, they obtain either sub-exponential (for sub-Gaussian tails) or polynomial decay rates toward equilibrium, with explicit dependencies on tail parameters $\beta$ or $\gamma$. Numerical simulations in one dimension corroborate the energy-structured form of the steady state, illustrating its approximate dependence on the energy $E(x,v)$ and matching the predicted tail profiles. This work extends previous results to a broader class of fat-tailed local equilibria and strengthens the connection between Harris-type methods and kinetic equations with non-explicit steady states.

Abstract

In this article, we prove some convergence results for kinetic Fokker-Planck equations with strong space confinement but fat-tailed local equilibria and non-explicit global steady states. We extend the results of \cite{C21} to a wider class of fat-tailed local equilibria, with rates of convergence in a large class of weighted $\sfL^1$ spaces. We complement our results with numerical simulations to investigate the shape of the non-explicit steady state.

Convergence to a non-explicit steady state in non-factorized kinetic Fokker-Planck equations with (very) weak velocity confinements

TL;DR

The paper proves existence and uniqueness of a non-explicit steady state for a kinetic Fokker-Planck equation with strong space confinement and fat-tailed local equilibria, and establishes convergence toward this state in weighted spaces. The authors develop weak Lyapunov functionals tailored to sub-Gaussian and polynomial tails and verify a positivity condition via kinetic Harnack inequalities, enabling a sub-geometric Harris-type convergence analysis. Depending on tail behavior, they obtain either sub-exponential (for sub-Gaussian tails) or polynomial decay rates toward equilibrium, with explicit dependencies on tail parameters or . Numerical simulations in one dimension corroborate the energy-structured form of the steady state, illustrating its approximate dependence on the energy and matching the predicted tail profiles. This work extends previous results to a broader class of fat-tailed local equilibria and strengthens the connection between Harris-type methods and kinetic equations with non-explicit steady states.

Abstract

In this article, we prove some convergence results for kinetic Fokker-Planck equations with strong space confinement but fat-tailed local equilibria and non-explicit global steady states. We extend the results of \cite{C21} to a wider class of fat-tailed local equilibria, with rates of convergence in a large class of weighted spaces. We complement our results with numerical simulations to investigate the shape of the non-explicit steady state.
Paper Structure (13 sections, 10 theorems, 101 equations, 4 figures)

This paper contains 13 sections, 10 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Weak Lyapunov functions
  3. Preliminaries
  4. Sub-Gaussian local equilibrium.
  5. Fat-tailed local equilibrium
  6. About the positivity condition
  7. Proofs of \ref{['thm:mainEXP']} and \ref{['thm:mainPOLY']}
  8. Numerical results
  9. Description of the algorithm
  10. Plot of the density of the steady state
  11. Level plots of the steady state
  12. Acknowledgments
  13. Reminder on the Harris theorem

Key Result

Theorem 1.1

When $\mathcal{M}$ is given by eq:Mexp with $\beta > 0$, there exists a positive normalised steady state $G\in \mathsf{L}^1\left(\exp({\delta E^{\frac{\min\{\beta,2\}}{2}}})\right)$ with $\delta>0$ small enough. Moreover, let $f$ be a solution to L, with initial data $f_0$. Then, for every $\theta\i

Figures (4)

  • Figure 5.1: Blue line: evolution of the density $\rho_f$ of a solution $f$ to kinetic Fokker-Planck equation with $\alpha = 1.5$, $\beta=0.5$ and $\delta =1.15$. Black dashed line: expected asymptotic behaviour \ref{['eq:asympRhoG']} with $\delta =1.15$.
  • Figure 5.2: Blue line: evolution the density of the kinetic Fokker-Planck equation with $\alpha = 1$, $\beta=1$ and $\delta = 2$. Black dashed line: expected asymptotic behaviour \ref{['eq:asympRhoG']} with $\delta=2$.
  • Figure 5.3: Profile and contour-plot of the solution $f$ of the kinetic Fokker-Planck equation at time $T=350$ for $\alpha = 1.5$ and $\beta=0.5$ and $\delta = 1.15$.
  • Figure 5.4: Profile and contour-plot of the steady state of the kinetic Fokker-Planck equation for $\alpha = 1$ and $\beta=1$ and $\delta = 2$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Lemma 2.1
  • proof : Proof of \ref{['lem:tech']}
  • Proposition 2.2
  • proof : Proof of \ref{['lem:H']}
  • Proposition 2.3
  • proof : Proof of \ref{['lem:Hpoly']}
  • Proposition 3.1
  • ...and 5 more