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Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations. (English. French summary) Zbl 1420.47026
In this interesting paper, the authors investigate the spectral analysis and long time asymptotic convergence of semigroups associated to discrete, fractional and classical Fokker-Planck equations. We recall that Fokker-Planck equations model time evolution of a density function of particles undergoing both diffusion and (harmonic) confinement mechanisms. They prove that the convergence is exponentially fast for a large class of initial data taken in a fixed weighted Lebesgue or weighted Sobolev space, with a rate of convergence which can be chosen uniformly with respect to the diffusion term. They investigate three regimes where these diffusion operators are close and for which such a uniform convergence can be established. The proofs are based on a deep and innovative splitting of the generator method.

##### MSC:
 47N20 Applications of operator theory to differential and integral equations 47D06 One-parameter semigroups and linear evolution equations 35B40 Asymptotic behavior of solutions to PDEs 35Q84 Fokker-Planck equations
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