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Pré-Publication, Document De Travail Année : 2015

Uniform semigroup spectral analysis of the discrete, fractional & classical Fokker-Planck equations

Résumé

In this paper, we investigate the spectral analysis (from the point of view of semi-groups) of discrete, fractional and classical Fokker-Planck equations. Discrete and fractional Fokker-Planck equations converge in some sense to the classical one. As a consequence, we first deal with discrete and classical Fokker-Planck equations in a same framework, proving uniform spectral estimates using a perturbation argument and an enlargement argument. Then, we do a similar analysis for fractional and classical Fokker-Planck equations using an argument of enlargement of the space in which the semigroup decays. We also handle another class of discrete Fokker-Planck equations which converge to the fractional Fokker-Planck one, we are also able to treat these equations in a same framework from the spectral analysis viewpoint, still with a semigroup approach and thanks to a perturbative argument combined with an enlargement one. Let us emphasize here that we improve the perturbative argument introduced in [7] and developed in [11], relaxing the hypothesis of the theorem, enlarging thus the class of operators which fulfills the assumptions required to apply it.
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Dates et versions

hal-01177101 , version 1 (16-07-2015)
hal-01177101 , version 2 (03-01-2016)
hal-01177101 , version 3 (02-03-2016)

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Stéphane Mischler, Isabelle Tristani. Uniform semigroup spectral analysis of the discrete, fractional & classical Fokker-Planck equations . 2015. ⟨hal-01177101v3⟩
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