发表机构
Bielefeld University; Rice University; Academy of Mathematics and System Sciences, CAS; The Chinese University of Hong Kong, Shenzhen（比勒费尔德大学; 莱斯大学; 中国科学院数学与系统科学研究院; 香港中文大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Nemytskii型Graphon McKean-Vlasov随机微分方程，证明弱解与强解的存在唯一性，并给出相关Fokker-Planck方程的新唯一性结果。
AI 中文摘要
我们研究了一个由一族本质上两两独立的Wiener过程驱动的、具有Nemytskii型系数的不可数系统的McKean-Vlasov随机微分方程。这些随机微分方程通过一个Graphon核相互作用，其相互作用依赖于在空间坐标上评估的一维时间边际律密度。我们在漂移系数和Graphon核的温和条件下证明了此类随机微分方程的依概率弱解的存在性和唯一性。此外，通过本质上证明Fubini扩展空间上的（受限）Yamada--Watanabe定理，我们证明了这些解实际上是依概率强解。对于相关的非线性Fokker-Planck方程组，我们证明了一个新的唯一性结果，其中我们允许Nemytskii型的密度依赖扩散系数。
英文摘要
We study an uncountable system of McKean-Vlasov SDEs with coefficients of Nemytskii-type which are driven by a family of essentially pairwise independent Wiener processes. These SDEs interact through a Graphon kernel by means of their one-dimensional time marginal law densities evaluated in the spatial coordinate. We prove the existence and uniqueness of probabilistically weak solutions to such SDEs under mild conditions on the drift coefficient and the Graphon kernel. Furthermore, we prove that these solutions are, in fact, probabilistically strong by essentially proving a (restricted) Yamada--Watanabe theorem on Fubini extension spaces. For the associated system of nonlinear Fokker-Planck equations, we prove a new uniqueness result where we can allow for density dependent diffusion coefficients of Nemytskii-type.
Comments38 pages, 1 figure
