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Exponential stability and hypoelliptic regularization for the kinetic Fokker-Planck equation with confining potential

Anton Arnold, Gayrat Toshpulatov

Abstract

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker-Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted $H^1$-norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted $L^2$-distance between a Fokker-Planck-solution and the steady state has always a sharp decay estimate of the order $\mathcal O\big( (1+t)e^{-tν/2}\big)$, with $ν$ the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker-Planck equations (from a weighted $L^2$-space to a weighted $H^1$-space).

Exponential stability and hypoelliptic regularization for the kinetic Fokker-Planck equation with confining potential

Abstract

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker-Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted -norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted -distance between a Fokker-Planck-solution and the steady state has always a sharp decay estimate of the order , with the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker-Planck equations (from a weighted -space to a weighted -space).
Paper Structure (12 sections, 11 theorems, 246 equations)

This paper contains 12 sections, 11 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Modified entropy methods for degenerate Fokker-Planck equations
  4. The choice of the matrix P
  5. Proofs
  6. Proof of Theorem \ref{['Main']}
  7. Proof of Proposition \ref{['prop.']}
  8. Proof of Theorem \ref{['Hyp.ellip.']} and Corollary \ref{['Cor.']}
  9. Appendix
  10. Proof that Assumption 2.\ref{['A2']}' implies Assumption \ref{['A1']}
  11. Matrix inequalities for Section 5.1
  12. Proof of inequality \ref{['aA']}

Key Result

Theorem 2.3

Let $V$ be a $C^{\infty}$ potential in $\mathbb{R}^n$ satisfying Assumptions Assum:Poincare and A1. Let $C_{PI},$$c,$$\tau,$ and $\alpha_0$ be the constants in Poincare, Condition1, and alpha-zero. Suppose the initial data $f_0$ satisfies $\frac{f_0}{f_{\infty}}\in H^1(\mathbb{R}^{2n}, f_{\infty})$ Then there are explicitly computable constants $C>0$ and $\lambda>0$ (independent of $f_0$) such th

Theorems & Definitions (30)

  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Remark 2.9
  • Example 2.10: Polynomial confining potentials
  • Remark 2.11
  • Theorem 3.1: NFP
  • ...and 20 more