Uniform semigroup spectral analysis of the discrete, fractional & classical Fokker-Planck equations
Mischler, Stéphane; Tristani, Isabelle (2015), Uniform semigroup spectral analysis of the discrete, fractional & classical Fokker-Planck equations. https://basepub.dauphine.fr/handle/123456789/17137
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-01177101
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
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Abstract (EN)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  and developed in , relaxing the hypothesis of the theorem, enlarging thus the class of operators which fulfills the assumptions required to apply it.
Subjects / Keywordssemigroup; dissipativity; Fokker-Planck equation; fractional Laplacian; spectral gap; exponential rate of convergence; long-time asymptotic
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