发表机构
Nankai University; CentraleSupélec, Université Paris-Saclay; Fudan University（南开大学; 巴黎萨克雷中央理工-高等学院; 复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究半离散二次Wasserstein能量的性质，构建Langevin温度的熵正则化松弛控制，证明相关动力学的适定性与遍历性，为该领域提供理论支撑。
AI 中文摘要
我们研究半离散二次Wasserstein能量，该能量在位点碰撞处非光滑。我们证明其在全构型空间上的局部Lipschitz连续性，以及全局半凹性、强制性和耗散性；证明每个全局极小值点均为内部点且无碰撞；并在无碰撞构型空间上建立C²正则性。梯度通过平衡Laguerre胞的重心表示，Hessian由显式面公式给出且满足全局单侧界。我们还在每个排序腔中显式求解一维问题，并给出单位正方形上具有非极小Lloyd不动点的两点示例。对于维度d≥2，我们随后构建Langevin温度的熵正则化松弛控制，受控动力学具有适定性、非爆炸性且无碰撞。其值函数是探索性Hamilton-Jacobi-Bellman方程的经典内部解；值函数的Laplacian局部为C¹，由此得到局部Lipschitz最优温度反馈。与该最优控制结果无关，对于每个固定的远离零的Borel温度规则，以及每个足够小的步长，相关高斯欧拉链是几何遍历的，具有全支撑的不变律。原始迭代不收敛，而当前最优能量几乎必然收敛到全局极小值，运行记录趋近于全局极小值集合。
英文摘要
We study the semi-discrete quadratic Wasserstein energy. The energy is nonsmooth at collisions of sites. We prove local Lipschitz continuity on the full configuration space, together with global semiconcavity, coercivity, and dissipativity; show that every global minimizer is interior and collision free; and establish $C^2$ regularity on the collision-free configuration space. The gradient is expressed through the barycenters of the balanced Laguerre cells, while the Hessian is given by an explicit facet formula and satisfies a global one-sided bound. We also solve the one-dimensional problem explicitly in each ordering chamber and give a two-site example on the unit square with non-minimizing Lloyd fixed points. For $d\ge 2$, we then formulate an entropy-regularized relaxed control of the Langevin temperature. The controlled dynamics is strongly well posed, nonexplosive, and collision free. Its value function is a classical interior solution of the exploratory Hamilton-Jacobi-Bellman equation; the Laplacian of the value function is locally $C^1$, which yields a locally Lipschitz optimal temperature feedback. Independently of this optimal-control result, for every fixed Borel temperature rule bounded away from zero, and every sufficiently small step size, the associated Gaussian Euler chain is geometrically ergodic with a full-support invariant law. The raw iterates do not converge, whereas the best-so-far energy converges almost surely to the global minimum and the running record approaches the set of global minimizers.
