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<p>We prove long-time contractivity estimates and exponential rate of convergence to equilibrium for solutions of hypoelliptic diffusion equations, which include the well-known Kolmogorov equation and similar kinetic Fokker–Planck equations in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\mathbb {R}}}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. 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-variable (coupling) methods. Next, we upgrade the estimate to weighted <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-(or total variation) norms, thanks to short-time hypocoercivity gradient estimates.</p>

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

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  • 发表日期：

    2026年5月2日

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  • Nicolò Forcillo,
  • Alessio Porretta

摘要

We prove long-time contractivity estimates and exponential rate of convergence to equilibrium for solutions of hypoelliptic diffusion equations, which include the well-known Kolmogorov equation and similar kinetic Fokker–Planck equations in \({{\mathbb {R}}}^d\) 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-variable (coupling) methods. Next, we upgrade the estimate to weighted \(L^1\) L 1 -(or total variation) norms, thanks to short-time hypocoercivity gradient estimates.