2025/07/31 by Luca Bondi, Bondi, Luca, Elena Issoglio +3
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Optical properties and cooling technologies in crystalline materials #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2507.23553
openalex publication_date 2025/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper focuses on recent works on McKean-Vlasov stochastic differential equations (SDEs) involving singular coefficients. After recalling the classical framework, we review existing recent literature depending on the type of singularities of the coefficients: on the one hand they satisfy some integrability and measurability conditions only, while on the other hand the drift is allowed to be a generalised function. Different types of dependencies on the law of the unknown and different noises will also be considered. McKean-Vlasov SDEs are closely related to non-linear Fokker-Planck equations that are satisfied by the law (or its density) of the unknown. These connections are often established also in this singular setting and will be reviewed here. Important tools for dealing with singular coefficients are also included in the paper, such as Figalli-Trevisan superposition principle, Zvonkin transformation, Markov marginal uniqueness, and stochastic sewing lemma.