TL;DR
This paper extends the Bakry-Émery method to hypocoercive diffusion operators, including the kinetic Fokker-Planck equation, providing a new framework to derive hypocoercive estimates.
Contribution
It introduces a generalized Bakry-Émery criterion applicable to Kolmogorov operators, unifying and extending Villani's hypocoercivity results.
Findings
Recovered hypocoercive estimates for the kinetic Fokker-Planck equation
Extended Bakry-Émery machinery to Kolmogorov operators
Provided a unified framework for hypocoercivity analysis
Abstract
We revisit Villani's approach to the study of hypocoercive diffusion operators by applying a variant of the Bakry-\'Emery machinery. The method relies on a generalized Bakry-\'Emery type criterion that applies to Kolmogorov type operators. Our approach includes as a special case the kinetic Fokker-Planck equation and allows, in that case, to recover hypocoercive estimates first obtained by Villani.
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