Convergence to equilibrium for the kinetic Fokker-Planck equation on the torus

TL;DR

The article studies convergence to equilibrium for the kinetic Fokker-Planck equation on the torus and proves exponential decay in the MKW distance using a non-Markovian coupling that leverages partial velocity noise to compensate for spatial spreading. It provides an explicit bound $\\mathcal{W}_2(\\mu_t,\\nu_t) \\le (e^{-\\lambda t} + c \\; e^{-t/(4\\lambda^2 L^2)})\\mathcal{W}_2(\\mu_0,\\nu_0)$ and shows that no universal contraction bound exists for all solutions. The paper also analyzes co-adapted (immersed) couplings, proving their existence with exponential decay but with weaker (sqrt) dependence on the initial distance, and demonstrates a fundamental sqrt lower bound $\\alpha(z) \\ge c \\sqrt{z}$ for initial separation, indicating intrinsic limitations of Markovian approaches. Together, these results advance understanding of hypocoercivity and provide quantitative, transport-based rates for convergence to equilibrium in kinetic models on compact manifolds.

Abstract

We study convergence to equilibrium for the kinetic Fokker-Planck equation on the torus. Solving the stochastic differential equation, we show exponential convergence in the Monge-Kantorovich-Wasserstein $\mathcal{W}_2$ distance. Finally, we investigate if such a coupling can be obtained by a co-adapted coupling, and show that then the bound must depend on the square root of the initial distance.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Definition 3: co-adapted coupling
• Theorem 4
• Remark 5
• Theorem 6
• Lemma 7
• proof
• Lemma 8
• proof