AI 中文总结
该研究探讨非光滑能量核生成的MMD的Wasserstein梯度流，证明不同维度与$q$值下的适定性、粒子系统收敛性及平衡态特性，还指出相关衰减模与不等式的不存在性。
AI 中文摘要
我们研究由非光滑能量核$K(z)=-|z|^q$（$0<q<2$）生成的平方最大均值差异（MMD）的Wasserstein梯度流。在维数$d\boldsymbol{\text{≥}}2$时，对应的能量不满足位移半凸性，因此标准的Wasserstein梯度流理论不适用。当$d+q-2>0$时，我们证明了$\boldsymbol{\text{R}}^d$上亚临界$L^p$空间中概率密度的整体适定性，其目标属于同一可积类且具有有限矩。我们还包含一维库仑端点情况$d=q=1$。对于相关的$N$粒子系统，我们证明了全局无碰撞、固定$N$收敛到无碰撞临界集、粒子到连续体的临界性原理，以及调制能量平均场估计，该估计表明当$N\to\boldsymbol{\text{∞}}$时，粒子动力学在每个有限时间区间上收敛到连续体流。我们还构造了无碰撞鞍点平衡，表明确定性粒子轨迹不一定趋近于全局经验极小值。对于$1\boldsymbol{\text{≤}}q<2$，我们类中的每个连续体解都具有窄相对紧轨道，每个$\boldsymbol{\text{ω}}$极限点都是拉格朗日临界点，且轨道趋近于拉格朗日临界集。对于$0<q<1$，在时间均匀矩和亚临界$L^p$界下，相同结论成立。我们证明当源和目标具有$q$阶有限矩时，绝对连续拉格朗日临界点等于目标，除了当$0<q<1$且$d\boldsymbol{\text{∈}}\boldsymbol{\text{\textit{\textlbrace1,3\textrbrace}}}$的情况。在上述均匀界下，刚性特性使得连续体流在适定性范围的刚性部分内收敛到目标。最后，我们证明在$\boldsymbol{\text{R}}^d$上不存在与初始数据无关的乘法MMD衰减模，并且在多个全空间和周期Riesz/库仑 regime中全局Polyak--Łojasiewicz不等式不成立。
英文摘要
We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels $K(z)=-|z|^q$, $0<q<2$. In dimensions $d\ge2$, the corresponding energies are not displacement semiconvex, so standard Wasserstein-gradient-flow theory does not apply. When $d+q-2>0$, we prove global well-posedness on $\mathbb{R}^d$ for probability densities in subcritical $L^p$ spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint $d=q=1$. For the associated $N$-particle system, we prove global noncollision and fixed-$N$ convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as $N\to\infty$ on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For $1\le q<2$, every continuum solution in our class has a narrowly relatively compact orbit, every $ω$-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For $0<q<1$, the same conclusions hold under uniform-in-time moment and subcritical $L^p$ bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order $q$, except when $0<q<1$ and $d\in\{1,3\}$. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on $\mathbb{R}^d$, and that global Polyak--Łojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.
