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<p>We study the rate of convergence to equilibrium of the solutions to Fokker–Planck type equations with linear drift by means of Cramér and Energy distances, which have been recently widely used in problems related to AI, in particular for tasks related to machine learning. In all cases in which the Fokker–Planck type equations can be treated through these distances, it is shown that the rate of decay is improved with respect to known results which are based on the decay of relative entropy.</p>

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ENERGY DISTANCE AND EVOLUTION PROBLEMS: A PROMISING TOOL FOR KINETIC EQUATIONS

  • 期刊论文

  • 发表日期：

    2026年2月19日

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  • Gennaro Auricchio,
  • Giuseppe Toscani

摘要

We study the rate of convergence to equilibrium of the solutions to Fokker–Planck type equations with linear drift by means of Cramér and Energy distances, which have been recently widely used in problems related to AI, in particular for tasks related to machine learning. In all cases in which the Fokker–Planck type equations can be treated through these distances, it is shown that the rate of decay is improved with respect to known results which are based on the decay of relative entropy.