Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness

2024/03/26 by Pascal Auscher, Cyril Imbert, Auscher, Pascal +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Mathematical Biology Tumor Growth #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2403.17464

openalex created_date 2024/03/26 · openalex publication_date 2024/03/26 · openalex updated_date 2026/08/01

Abstract

We prove existence, uniqueness and regularity of weak solutions of Kolmogorov--Fokker--Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the Cauchy problem and allow a broad class of source terms under minimal assumptions. The core of the analysis is a set of sharp kinetic embeddings à la Lions and transfer-of-regularity results à la Bouchut--H''ormander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.

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• Nash's G bound for the Kolmogorov equation

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