AI 中文总结
研究N粒子随机最优控制问题值函数到平均场控制问题值函数的收敛速率，通过构建重耦合影子流证明d≥2时速率最优，消除半凹性假设，还发现一维中合作粒子有更好表现及最优多项式指数。
AI 中文摘要
我们证明，对于仅在1-瓦瑟斯坦距离上为利普希茨连续的平均场成本，N粒子随机最优控制问题的值函数以经验测度速率（d≥3时为N^(-1/d)，d = 2时为N^(-1/2)√(log N)）一致收敛到相应平均场控制问题的值函数。对于d≥2，该速率在这类问题中是最优的，这类问题的平均场优化器既不唯一也不稳定。这证明了Daudin、Delarue和Jackson所推测的速率，并消除了他们提出的半凹性假设。证明是控制理论性的：从N粒子控制的每个实现中构建一个路径依赖的福克-普朗克流——一个影子流，通过最优传输反复与粒子重耦合，以经验测度速率跟踪经验测度。在存在加性共同噪声时，相同的速率在其强度上一致成立。最后，在一维中我们发现经验测度基准不是最优的：合作粒子能超越它，最优多项式指数是4/7，严格介于独立样本的精度和自由放置点的量化精度之间。证明结合了影子流的预期校正、实现合作的吉布斯定律以及最优性的薛定谔基态估计。
英文摘要
We determine the rate of convergence of the value functions of the $N$-particle stochastic optimal control problem to the value function of the corresponding mean field control problem, for mean field costs that are merely Lipschitz continuous in the 1-Wasserstein distance, a class that covers problems whose mean field optimizers are neither unique nor stable. For $d\geq2$, the optimal rate is the empirical-measure rate ($N^{-1/d}$ for $d\geq3$, $N^{-1/2}\sqrt{\log N}$ for $d=2$): this proves the rate conjectured by Daudin, Delarue and Jackson, and removes the semiconcavity hypothesis made there. In dimension one, we discover that the empirical-measure benchmark is not optimal: cooperating particles beat it, and the optimal polynomial exponent is $4/7$, strictly between the accuracy of independent samples and that of quantization by freely placed points. The proofs are control-theoretic: from each realization of an $N$-particle control we build a pathwise Fokker-Planck flow (a "shadow flow") which, repeatedly recoupled to the particles by optimal transport, shadows the empirical measure at the optimal rate. The one-dimensional rate requires additional constructions: we correct the shadow flow with a filter built on the future of the discarded noise, draw the cooperating particles from a Gibbs law, and prove the optimality of the exponent by a Schrodinger ground-state estimate. The empirical-measure rate also holds under additive common noise, uniformly in its intensity.
Commentsv2: Expanded and improved exposition in Section 6; the future filter construction is now motivated in detail. Minor edits elsewhere. Results unchanged
