AI 中文总结
研究概率测度与目标测度间平方MMD的瓦瑟斯坦梯度流长期行为，通过建立全局弱解、证明超压缩估计等研究其正则性，在不同空间证明了相关不等式及收敛性，确定空间无穷远处障碍，给出特定条件下指数收敛结果。
AI 中文摘要
我们研究了概率测度\(\rho\)与目标测度\(\mu\)之间平方最大均值差异（MMD）的瓦瑟斯坦梯度流的长期行为，其基础核由库仑势给出。首先，我们建立了从任意博雷尔概率测度出发的全局弱解的存在性，并证明了一个超压缩估计，表明对于\(t>0\)，密度\(\rho_t\)在\(L^\infty\)中立即有界。我们还研究了这些解的正则性，表明赫尔德范数可以随时间指数增长。其次，在平坦环面\(\mathbb{T}^{\mathsf{d}}\)上，我们证明了沿着流向均匀正目标\(\mu\)的平方MMD的指数衰减，而无需对初始数据有下界。该结果基于一个“缺陷Polyak - Lojasiewicz（PL）不等式”，其缺陷项考虑了演化密度中可能的真空区域。我们还证明了当目标仅在一点消失时，通常的PL不等式可能不成立，并且在至少二维的情况下，没有强制常数可以仅依赖于目标的规定正下界。最后，在\(\mathbb{R}^{\mathsf{d}}\)上，我们确定了空间无穷远处的一个障碍。对于紧支集目标，最初与目标相距距离\(D\)的均匀局部化源在时间量级为\(D\)时保留其初始平方MMD的固定比例。因此，在无限制的全空间类上，既不能有在初始数据上均匀的乘性平方MMD衰减模量，也不能有全局PL不等式。相比之下，在径向对称、源支持包含和目标正性假设下，我们建立了一个PL不等式和指数收敛。
英文摘要
We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $ρ$ and a target measure $μ$, where the underlying kernel is given by a Coulomb potential. For $L^\infty$ target densities $μ$, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove that the density $ρ_t$ belongs to $L^\infty$ for any $t>0$. We also show that the Hölder norm can grow exponentially in time. On the flat torus ${\mathbb{T}}^\mathsf{d}$, we prove a global metric PL inequality for every finite-Coulomb-energy source and nearly uniform target. For general bounded, uniformly positive targets, we prove exponential decay of the squared MMD without requiring a lower bound on the initial data, using a defective PL inequality. We also prove that the usual PL inequality may fail when the target vanishes only at one point and that, when $\mathsf{d}\ge2$, no PL constant can hold uniformly over all targets satisfying a prescribed lower bound. On ${\mathbb{R}}^\mathsf{d}$, for $\mathsf{d}\ge2$, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence. On the unrestricted whole-space class, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold. Finally, in every dimension and in both spatial settings, we prove that every Lagrangian critical point coincides with the target when $(ρ-μ)^+$ is absolutely continuous. In dimension two, the energy supplies uniform tightness. This implies that if our constructed solutions have finite energy at some positive time, then they converge to the target narrowly and strongly in negative-order Sobolev spaces.
Comments50 pages, v2: (i) Added a global metric PL inequality on the torus for near-uniform targets, with the implied exponential convergence for the gradient flows. (ii) Proved rigidity of Lagrangian critical points when the positive part of the source-target discrepancy is absolutely continuous, and obtained qualitative convergence of finite-energy solutions on R^2 via logarithmic-capacity tightness
