Regularization of the superposition principle: Potential theory meets Fokker-Planck equations

[2603.05174] Regularization of the superposition principle: Potential theory meets Fokker-Planck equations Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2603.05174 (math) [Submitted on 5 Mar 2026 ( v1 ), last revised 24 Aug 2026 (this version, v3)] Title: Regularization of the superposition principle: Potential theory meets Fokker-Planck equations Authors: Lucian Beznea , Iulian Cîmpean , Michael Röckner View PDF HTML (experimental) Abstract: For a solution to a (possibly nonlinear) Fokker-Planck equation (FPE) the powerful superposition principle renders a probability measure on path space with one dimensional time marginals equal to this solution, and additionally solving the martingale problem for the Kolmogorov operator given by the FPE. The superposition principle thus reveals that such parabolic PDEs have a probabilistic counter part. The aim of this work is to go a substantial further step and, by exploiting the superposition principle, construct a full fledged Markov process, i.e. a family of path space measures for a large set of space time starting points connected by the Markov property, associated to the (linearized) FPE in the above way. Under very general (merely measurability) conditions on the coefficients of the FPE this is achieved in this paper in such a way that the resulting process is a right process, which is a particularly useful class of Markov processes, enjoying among other regularity properties the strong Markov property, which is fundamental for the analysis of the underlying FPE as a (nonlinear) parabolic PDE by probabilistic tools. As two main applications we construct fundamental flow solutions for the FPE and we prove a well-posedeness result for the parabolic Dirichlet problem through probabilistic means for more general coefficients than could be treated in the existing literature. Furthermore, we introduce a Choquet capacity for such FPEs using the corresponding right process. The validity of the strong Markov property in the context of the superposition principle was an open problem even in the linear case. In this paper we solve this also in the nonlinear case, i.e. for path laws of solutions to McKean-Vlasov SDEs with Nemytskii type coefficients. A main application here is the FPE given by the generalized porous media equation and its corresponding McKean-Vlasov SDE. Subjects: Probability (math.PR) ; Analysis of PDEs (math.AP) Cite as: arXiv:2603.05174 [math.PR] (or arXiv:2603.05174v3 [math.PR] for this version) https://doi.org/10.48550/arXiv.2603.05174 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Iulian Cimpean [ view email ] [v1] Thu, 5 Mar 2026 13:42:03 UTC (94 KB) [v2] Wed, 11 Mar 2026 18:10:51 UTC (94 KB) [v3] Mon, 24 Aug 2026 19:38:19 UTC (95 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2026-03 Change to browse by: math math.AP 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) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

Related documents

• Regularization of the superposition principle: Potential theory meets Fokker-Planck equations#670513
• Existence and Blow-Up for Non-linear Fokker-Planck with Controlled Drift#1061396
• Integral regularization PINNs for evolution equations#158685
• Absence of free-time minimizers for N-body problems and a splitting theorem#1155894
• A Fourier Neural Operator for Accelerated Discretization-Invariant Solutions of the Radiative Transfer Equation#662501