发表机构
Southern University of Science and Technology; Univ. Rennes, CNRS, IRMAR - UMR 6625（南方科技大学; 雷恩大学，法国国家科学研究中心，雷恩数学与随机过程研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究平均场正倒向随机微分方程的混沌传播，在单调性和Lipschitz条件下获得尖锐的Wasserstein误差估计，并建立弱误差转移原理，为倒向分量提供匹配下界。
AI 中文摘要
我们研究了解耦的平均场正倒向随机微分方程的混沌传播，其中生成元依赖于前向状态、倒向值和对角鞅被积函数的经验分布。在单调性和Lipschitz假设下，同步耦合给出了定量估计，包括对于通过有限多个统计量相互作用的n个粒子系统，其m粒子平方Wasserstein界为m/n阶。在马尔可夫设定下，假设存在足够正则的经典解耦场，我们获得了两个尖锐的改进。对于常数可逆扩散以及沿极限律流场的测度依赖的一阶抵消，平方Wasserstein误差在连续路径空间上的值和对角被积函数的L^2空间中均为m^2/n^2阶。在不施加此抵消的情况下，对于每个固定边际，光滑弱误差具有n^{-1}阶，允许可变且可能退化的扩散。弱估计在固定时间区间上对值是一致成立的，而对被积函数则是时间积分的。该论证将相互作用的BSDE与解耦场的经验评估进行比较，保留了完整的鞅表示，并通过两个倒向律控制反馈。它产生了一个具有固有平方误差m/n^2的联合路径Wasserstein转移界、非对角被积函数估计，以及一个具有加性误差n^{-1}的弱误差转移原理。通过两个倒向律反馈的显式模型验证了抵消假设。例子区分了固有倒向误差与前向律误差，并为每个倒向分量建立了匹配的下界。
英文摘要
We study propagation of chaos for decoupled mean-field forward-backward stochastic differential equations whose generators depend on the empirical laws of the forward states, backward values and diagonal martingale integrands. Under monotonicity and Lipschitz assumptions, synchronous coupling gives quantitative estimates, including an $m$-particle squared Wasserstein bound of order $m/n$ for a system of $n$ particles interacting through finitely many statistics. In the Markovian setting, assuming a sufficiently regular classical decoupling field, we obtain two sharp refinements. For constant invertible diffusion and first-order cancellation of the field's measure dependence along the limiting law flow, the squared Wasserstein error is of order $m^2/n^2$, on continuous-path space for the values and on $L^2$ for the diagonal integrands. Without imposing this cancellation, smooth weak errors have order $n^{-1}$ for every fixed marginal, allowing variable and possibly degenerate diffusion. The weak estimate is uniform on a fixed time interval for the values and integrated in time for the integrands. The argument compares the interacting BSDE with an empirical evaluation of the decoupling field, retaining the full martingale representation and controlling feedback through both backward laws. It yields a joint-path Wasserstein transfer bound with intrinsic squared error $m/n^2$, off-diagonal integrand estimates, and a weak-error transfer principle with additive error $n^{-1}$. Explicit models with feedback through both backward laws verify the cancellation assumptions. Examples distinguish the intrinsic backward error from the forward law error and establish matching lower bounds for each backward component.
Comments55 pages. This paper was written entirely by the authors. AI-assisted tools were used only for language polishing and grammatical correction, and not for generating scientific content
