New perspectives on recent trends for Kolmogorov operators

2023/04/03 by Francesca Anceschi, Mirco Piccinini, Anceschi, Francesca +3
Mathematics · Physics and Astronomy · #35B45 #35B65 #35K70 #35Q84 #35R11 #47G20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2304.01147

openalex publication_date 2023/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

After carrying out an overview on the non Euclidean geometrical setting suitable for the study of Kolmogorov operators with rough coefficients, we list some properties of the functional space W, mirroring the classical H¹ theory for uniformly elliptic operators. Then we provide the reader with the proof of a new Sobolev embedding for functions in W. Additionally, after reviewing recent results regarding weak regularity theory, we discuss some of their recent applications to real life problems arising both in Physics and in Economics. Finally, we conclude our analysis stating some recent results regarding the study of nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck operators.

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