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

[2510.07542] First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2510.07542 (math) [Submitted on 8 Oct 2025 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation Authors: Alessandro Pinzi View PDF HTML (experimental) Abstract: The goal of this paper is to define an evolution equation for a curve of random probability measures $(M_t)_{t\in[0,T]}\subset \mathcal{P}(\mathcal{P}(\mathbb{R}^d))$ associated to a non-local drift $b:[0,T]\times\mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$ and a non-local diffusion term $a:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \operatorname{Sym}_+(\mathbb{R}^{d\times d})$. Then, we show that any solution to such an equation on random measures can be lifted twice: to a superposition of solutions to a non-linear Kolmogorov-Fokker-Planck equation and to a superposition of weak solutions to the McKean-Vlasov equations. Finally, we exploit this nested 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 version of KFP. As a tool, we will introduce integral metrics over the spaces of probability measures $\mathcal{P}(\mathbb{R}^d)$ in duality with smooth functions, including a weighted second-order metric. Comments: 29 pages. Comments are welcome Subjects: Analysis of PDEs (math.AP) ; Functional Analysis (math.FA); Probability (math.PR) MSC classes: 60G57, 35R15, 35Q84, 60H15 Cite as: arXiv:2510.07542 [math.AP] (or arXiv:2510.07542v2 [math.AP] for this version) https://doi.org/10.48550/arXiv.2510.07542 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alessandro Pinzi [ view email ] [v1] Wed, 8 Oct 2025 20:57:45 UTC (45 KB) [v2] Thu, 27 Aug 2026 20:36:03 UTC (48 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2025-10 Change to browse by: math math.FA math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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