2015/07/17 by Stéphane Mischler, Mischler, Stéphane, Isabelle Tristani +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #advanced mathematical theories
paper · doi:10.48550/arxiv.1507.04861
openalex publication_date 2015/07/17 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28
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.