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Long-time contractivity estimates for kinetic Kolmogorov-Fokker-Planck equations

Nicolò Forcillo, Alessio Porretta

TL;DR

This work addresses the problem of long-time convergence for hypoelliptic kinetic Kolmogorov-Fokker-Planck equations by introducing a self-contained PDE approach that derives exponential contraction via dual oscillation estimates on the adjoint problem. The authors first prove Wasserstein contraction through a doubling-variables coupling argument and then upgrade to weighted total-variation decay using short-time $L^{\infty}$-to-$W^{1,\infty}$ smoothing (hypocoercivity). The key contributions include (i) a rigorous oscillation-decay framework for the adjoint problem leading to $W_1$-type contraction, (ii) a Bernstein-type short-time smoothing result giving $L^{\infty}$-Lipschitz regularization, and (iii) explicit Lyapunov-function-based conditions yielding exponential decay in the weighted TV norm, with concrete examples such as the Kolmogorov equation. The results provide a robust, duality-based method that extends to nonlinear, inhomogeneous first-order terms and highlights the role of hypoellipticity in driving long-time stabilization without requiring explicit steady states.

Abstract

We prove long-time contractivity estimates and exponential rates of convergence to equilibrium for solutions of hypoelliptic diffusion equations, which include the well-known Kolmogorov equation and similar kinetic Fokker-Planck equations in $\R^d$. Compared to the existing literature, our proof exploits a different approach, elementary and self-contained, based on oscillation estimates for the adjoint problem. We first prove contractivity in Wasserstein distances through doubling variables (coupling) methods. Next, we upgrade the estimate to weighted $L^1$-(or total variation) norms, thanks to short-time hypocoercivity gradient estimates.

Long-time contractivity estimates for kinetic Kolmogorov-Fokker-Planck equations

TL;DR

This work addresses the problem of long-time convergence for hypoelliptic kinetic Kolmogorov-Fokker-Planck equations by introducing a self-contained PDE approach that derives exponential contraction via dual oscillation estimates on the adjoint problem. The authors first prove Wasserstein contraction through a doubling-variables coupling argument and then upgrade to weighted total-variation decay using short-time -to- smoothing (hypocoercivity). The key contributions include (i) a rigorous oscillation-decay framework for the adjoint problem leading to -type contraction, (ii) a Bernstein-type short-time smoothing result giving -Lipschitz regularization, and (iii) explicit Lyapunov-function-based conditions yielding exponential decay in the weighted TV norm, with concrete examples such as the Kolmogorov equation. The results provide a robust, duality-based method that extends to nonlinear, inhomogeneous first-order terms and highlights the role of hypoellipticity in driving long-time stabilization without requiring explicit steady states.

Abstract

We prove long-time contractivity estimates and exponential rates of convergence to equilibrium for solutions of hypoelliptic diffusion equations, which include the well-known Kolmogorov equation and similar kinetic Fokker-Planck equations in . Compared to the existing literature, our proof exploits a different approach, elementary and self-contained, based on oscillation estimates for the adjoint problem. We first prove contractivity in Wasserstein distances through doubling variables (coupling) methods. Next, we upgrade the estimate to weighted -(or total variation) norms, thanks to short-time hypocoercivity gradient estimates.

Paper Structure

This paper contains 11 sections, 16 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Notation
  4. Assumptions and well-posedness of the differential problems
  5. Oscillation decay estimates
  6. Exponential decay of oscillation seminorms
  7. Decay of Wasserstein distances for the kinetic Fokker-Planck equation
  8. Hypocoercivity and $\mathbf {L^\infty}$-Lipschitz smoothing
  9. Improved long-time decay
  10. Examples
  11. Appendix

Key Result

Theorem 1.1

Assume that $H,B$ satisfy (Blip) and (H1)-(H2), and there exists a Lyapunov function $\varphi$ (cf. Definition def-Lyap) fulfilling conditions (liplocvfi), (vfi2), and (vfi3). Then, there exist $\omega, K>0$ such that, for every initial data $m_{01}, m_{02}\in {\mathcal{P}}_1({\mathbb R}^{2d})$ for where $\|\cdot\|_{TV_\varphi}$ is the norm of the total variation weighted by $\varphi$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Corollary 2.5
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • ...and 25 more