Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations
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.
scientific article