流形上随机系统的Wasserstein收缩：一种微分方法
Wasserstein Contraction of Stochastic Systems on Manifolds: A Differential Approach
AI总结：
本文提出一种微分方法，通过均方微分构造随机变分动力学，实现流形上随机系统的Wasserstein收缩，并引入正交微分耦合，应用于随机控制以克服确定性全局稳定障碍。
AI中文摘要：
动力系统的收缩分析将全局收敛问题转化为局部问题。如果变分系统在一致意义下是稳定的，那么原始系统的所有轨迹都会相互收敛。本文为连续时间随机系统发展了一个随机类比，其中收敛性通过概率定律之间的Wasserstein距离来衡量。关键步骤是利用均方微分来构造随机变分动力学。其一致$L_2$稳定性随后可沿初始条件曲线积分，从而得到相关概率定律的Wasserstein收缩。这产生了以微分Lyapunov函数形式表述的准则，并且在流形上，它导出了内在的黎曼表述。随机设置的一个显著特征是噪声的耦合成为设计自由度。为了利用这一自由度，我们引入了一种正交微分耦合，将框架扩展到同步耦合之外。由此产生的准则明确揭示了状态空间的曲率如何进入收缩机制，并且在标准正交标架情形下，恢复了经典的Ricci曲率结构。最后，我们将该框架应用于紧流形上的随机控制。通过引入随机控制输入，我们克服了全局稳定的确定性障碍。也就是说，一个无法通过光滑时不变确定性反馈全局稳定的系统，在$W_2$中变为全局收缩，并且其稳态分布可以任意集中在规定的目标平衡点附近。
英文摘要：
Contraction analysis of dynamical systems turns a global convergence problem into a local one. If the variational system is stable in a uniform sense, then all trajectories of the original system converge toward each other. This paper develops a stochastic analogue for continuous-time stochastic systems, where convergence is measured by Wasserstein distance between probability laws. The key step is to employ mean-square differentiation to construct stochastic variational dynamics. Its uniform $L_2$ stability can then be integrated along curves of initial conditions, yielding Wasserstein contraction of the associated probability laws. This yields criteria formulated in terms of differential Lyapunov functions, and on manifolds, it leads to intrinsic Riemannian formulations. A distinctive feature of the stochastic setting is that the coupling of the noises becomes a design freedom. To exploit this freedom, we introduce an orthogonal differential coupling that extends the framework beyond synchronous coupling. The resulting criterion explicitly exposes how the curvature of the state space enters the contraction mechanism and, in the orthonormal-frame case, recovers the classical Ricci-curvature structure. Finally, we apply the framework to stochastic control on a compact manifold. By introducing stochastic control input, we overcome a deterministic obstruction to global stabilization. That is, a system that cannot be globally stabilized through smooth time-invariant deterministic feedback is rendered globally contracting in $W_2$, and its steady state distribution can be made arbitrarily concentrated around a prescribed target equilibrium.
