发表机构
Mathematical Institute, University of Oxford（牛津大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对有界域中反射扩散的边际分布流，给出密度/通量对存在正则拉格朗日流的充分条件，构造反例说明边界假设不可放松，还建立无通量Fokker-Planck方程的唯一性结果，为反射扩散模型的ODE采样提供数学依据。
AI 中文摘要
受有界域中反射扩散的边际分布流的启发，我们研究求解无通量连续性方程的密度/通量对何时能保持在闭域内并生成给定密度流的正则拉格朗日流。我们基于内部有界变差正则性、边界套层上的有界变差控制、绝对连续散度的单侧界以及速度的消失法向迹，给出了充分条件。证明利用了相切性消除零延拓散度的奇异边界贡献，从而使延拓后的速度可用于Ambrosio-DiPerna-Lions理论。我们证明这些边界假设无法联合放松以纳入边界流机制。我们构造了一个携带边界流的显式光滑密度/通量对，其密度演化在加权类中是唯一的，其特征是唯一、受限且传输边际，但它不承认正则拉格朗日流，因为在初始时刻附近可任意破坏可压缩性界。我们还建立了无通量Fokker-Planck方程的两个唯一性结果：有界可测漂移的对偶结果，以及在边界处奇异的入口型漂移的加权能量结果。我们的结果为在系数的最小正则性假设下使用基于ODE的反射扩散模型采样提供了严格的数学证明，也指明了此类基于ODE的采样器可能失效的情形。
英文摘要
Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density/flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. We give sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio-DiPerna-Lions theory. We show that these boundary assumptions cannot be jointly relaxed so as to admit a boundary current mechanism. We construct an explicit smooth density/flux pair carrying a boundary current. Its density evolution is unique in a weighted class and its characteristics are unique, confined and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. We also establish two uniqueness results for no-flux Fokker-Planck equations: a duality result for bounded measurable drifts and a weighted energy result for entrance-type drifts singular at the boundary. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.
Comments31 pages, 1 figure
