发表机构
Al.I. Cuza University; Octav Mayer Institute of Mathematics of Romanian Academy; Faculty of Mathematics, Bielefeld University; Academy for Mathematics and Systems Science, CAS; School of Data Science, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（阿尔·伊·库扎大学; 罗马尼亚科学院奥克塔夫·迈尔数学研究所; 比勒费尔德大学数学学院; 中国科学院数学与系统科学研究院; 香港中文大学（深圳）数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究具有非对角扩散项的非线性福克 - 普朗克方程分布解唯一性，证明温和解在更大类中唯一，扩展以往结果，还证相关线性化方程唯一性，应用于麦克凯恩 - 弗拉索夫随机微分方程，建立\(L^{\infty}\)估计并证明其解的路径律构成非线性马尔可夫过程。
AI 中文摘要
本文研究具有非对角扩散项的非线性福克 - 普朗克方程分布解的唯一性。在适当假设下，该方程生成连续收缩半群，\(u(t)=S(t)u_0\)是其温和解。主要贡献是证明此温和解在更大的分布解类中唯一，扩展了对角扩散情形的唯一性结果。还证明了相关线性化方程分布解的唯一性，应用于证明相应麦克凯恩 - 弗拉索夫随机微分方程的弱唯一性，建立了新的\(L^{\infty}\)估计并用于构造非线性马尔可夫过程，最后证明了麦克凯恩 - 弗拉索夫随机微分方程解的路径律构成非线性马尔可夫过程。
英文摘要
This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type \begin{equation} u_{t}-\sum_{i,j=1}^{d} D^{2}_{ij}(a_{ij}(x)β(x,u))+ \text{div}(b(x,u)u)=0 \quad \text{in}\; (0, \infty) \times \mathbb{R}^{d} ,\notag \end{equation} with initial condition $u(0,x)\equiv u_{0}(x)$, where $a_{ij}$, $β$, and $b$ are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup $S(t): L^{1}(\mathbb{R}^{d}) \rightarrow L^{1}(\mathbb{R}^{d})$, and $u(t)=S(t)u_{0}$ is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case $a_{ij} \equiv δ_{ij}$. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new $L^{\infty}$ estimate for mild solutions starting from data in $L^{1}\cap L^{\infty}$ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.
