Strong solutions to McKean-Vlasov SDEs associated to a class of degenerate Fokker-Planck equations with coefficients of Nemytskii-type

• 成果类型: Article; Early Access
• 署名作者: Grube, Sebastian
• 署名单位: University of Bielefeld
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01428-y
• 发表日期: 2025
• 关键词: distribution dependent sdes; MARKOV-PROCESSES; fluctuations; propagation; REGULARITY; chaos

摘要

While the nondegenerate case is well known, there are only few results on the existence of strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type in the degenerate case. We consider a broad class of degenerate nonlinear Fokker-Planck(-Kolmogorov) equations with coefficients of Nemytskii-type. This includes, in particular, the classical porous medium equation perturbed by a first-order term with initial datum in a subset of probability densities, which is dense with respect to the topology inherited from L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>1$$\end{document}, and, in the one-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point x0 is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0\in {\mathbb {R}}$$\end{document}. For these kind of equations the existence of a Schwartz-distributional solution u is well-known. We show that there exists a unique strong solution to the associated degenerate McKean-Vlasov SDE with time marginal law densities u. In particular, every weak solution to this equation with time marginal law densities u can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional yields a weak solution with time marginal law densities u.