First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation

2025/10/08 by Alessandro Pinzi, Pinzi, Alessandro
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35Q84 #35R15 #60G57 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Gas Dynamics and Kinetic Theory #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2510.07542

openalex publication_date 2025/10/08 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

The goal of this paper is to define an evolution equation for a curve of random probability measures (M_t)_t∈[0,T]⊂ P(P(ℝ^d)) associated to a non-local drift b:[0,T]×ℝ^d × P(ℝ^d) → ℝ^d and a non-local diffusion term a:[0,T]× ℝ^d × P(ℝ^d) → Sym₊(ℝ^{d× d}). Then, we show that any solution to that equation can be lifted to a superposition of solutions to a non-linear Kolmogorov-Fokker-Planck equation and also to a superposition of weak solutions to the McKean-Vlasov equations. Finally, we use this superposition result to show how existence and uniqueness can be transferred from the equation on random measures to the associated non-linear Kolmogorov-Fokker-Planck equation and to the McKean-Vlasov equation, assuming uniqueness of the linearized KFP.

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