发表机构
The University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带非局部相互作用的路径空间动态最优输运，证明高斯势的程函性质并推导出刘维尔型动力学方程，建立了端点几何与平均场动力学的变分联系。
AI 中文摘要
我们研究了路径空间上具有动力学代价和非局部相互作用的动态最优输运问题。对于高斯相互作用核，循环单调性和有效端点代价的正则性产生了一个与最优轨迹初始动量相关联的保守向量场。另外，高斯相互作用势 $W_{\pi_0}$ 满足一致梯度估计，并且在相互作用强度的定量条件下，它是程函方程的经典粘性下解。我们还通过保持端点的扰动直接由路径空间最优性推导出欧拉-拉格朗日动力学。相应的相空间边际满足由光滑高斯自洽力驱动的测度值刘维尔型或非局部动力学方程。反之，叠加原理将相空间连续性方程的合适测度值解提升为相空间轨迹上的测度。因此，路径空间最优输运在端点几何、高斯相互作用势的程函行为与动力学平均场动力学之间提供了变分联系。
英文摘要
We study a dynamical optimal transport problem on path-space with kinetic cost and nonlocal interaction. For a Gaussian interaction kernel, cyclical monotonicity and the regularity of the effective endpoint cost yield a conservative vector field associated with the initial momentum of the optimal trajectories. Separately, the Gaussian interaction potential $W_{π_0}$ satisfies a uniform gradient estimate and, under a quantitative condition on the interaction strength, is a classical and hence viscosity subsolution of an eikonal equation. We also derive the Euler--Lagrange dynamics directly from path-space optimality by means of endpoint-preserving perturbations of the optimal path measure. The associated phase-space marginals satisfy a measure-valued Liouville-type, or nonlocal kinetic, equation driven by the smooth Gaussian self-consistent force. Conversely, a superposition principle lifts suitable measure-valued solutions of the phase-space continuity equation to measures on phase-space trajectories. Thus path-space optimal transport provides a variational connection between endpoint geometry, eikonal behavior of the Gaussian interaction potential, and kinetic mean-field dynamics.
Comments37 pages, 1 figure, comments welcome
