发表机构
University of Connecticut; Huazhong University of Science and Technology（康涅狄格大学; 华中科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为多尺度平均场正倒向随机微分方程建立平均化原理，提出随机环境中的条件不变测度与均匀重启稳定性条件，证明最优收敛率，并应用于平均场随机控制问题。
AI 中文摘要
我们为一类多尺度平均场正倒向随机微分方程建立了平均化原理，并识别出若干在经典快慢系统中不存在的全新现象。与经典快慢系统相比，有效动力学通常不能通过简单冻结确定性慢参数并对所得快方程的平稳测度取平均来获得。合适的平均化对象是由随机环境中的冻结快动力学及其相关的条件不变测度提供的，这些测度保留了慢状态与其分布之间的耦合。正倒向结构造成了进一步的障碍：局部平均化估计在任意时间范围内传播时不一定保持稳定。我们为平均化系统确定了一个均匀重启稳定性条件，在该条件下这一障碍可以被克服。利用状态-律动力学的联合提升半群，结合双尺度离散化和Gordin型分解，我们证明了正倒向分量的强平均化，且收敛率达到最优阶$O(\varepsilon^{1/2})$。作为应用，我们将该一般理论应用于一类平均场随机控制问题，并开发了一种求解此类平均场控制问题的高效算法。
英文摘要
We establish an averaging principle for a class of multiscale mean-field forward-backward stochastic differential equations and identify several novel phenomena that are absent from classical fast-slow systems. In contrast with classical fast-slow systems, the effective dynamics cannot in general be obtained by simply freezing deterministic slow parameters and averaging against the invariant measure of the resulting fast equation. The appropriate averaging object is instead provided by a frozen fast dynamics in a random environment and its associated conditional invariant measures, which retain the coupling between the slow state and its distribution. The forward-backward structure creates a further obstruction: local averaging estimates need not remain stable when propagated over an arbitrary time horizon. We identify a uniform restart stability condition for the averaged system under which this obstruction can be overcome. Using a joint lifted semigroup for the state-law dynamics, together with a two-scale discretization and a Gordin-type decomposition, we prove strong averaging for both the forward and backward components with optimal convergence rate $O(\varepsilon^{1/2})$. As an application, we apply the general theory to a class of mean-field stochastic control problems and develop an efficient algorithm for solving such mean-field control problems.
Comments65 pages, 2 figures
