We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$ towards given target measures $\nu$ on the real line, where we focus on the negative distance kernel $K(x,y) := -|x-y|$. In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone $\mathcal C(0,1) \subset L_2(0,1)$ of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on $L_2(0,1)$. Based on the construction of an appropriate counterpart of $\mathcal F_\nu$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $\nu$, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of $\mathcal C(0,1)$. For certain $\mathcal F_\nu$-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme, which is easily computable by a bisection algorithm. For continuous targets $\nu$, also the explicit Euler scheme can be employed, although with limited convergence guarantees.

翻译：本文全面描述了实线上针对给定目标测度ν的最大均值差异(MMD)泛函$\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$的Wasserstein梯度流，其中我们聚焦于负距离核$K(x,y) := -|x-y|$。在一维情形下，Wasserstein-2空间可以等距嵌入到分位数函数构成的锥$\mathcal C(0,1) \subset L_2(0,1)$中，这导致通过$L_2(0,1)$上关联柯西问题的解来刻画Wasserstein梯度流。基于在$L_2(0,1)$上构造$\mathcal F_\nu$的适当对应泛函及其次微分，我们给出了该柯西问题的解。对于离散目标测度ν，这导出了分段线性的显式解公式。我们证明了该流在$\mathcal C(0,1)$子集上的不变性与平滑性质。对于某些$\mathcal F_\nu$-流，这意味着初始点测度会立即变为绝对连续测度，并随时间保持该性质。最后，我们通过采用隐式欧拉格式的各种数值算例说明了该流的行为，该格式可通过二分算法轻松计算。对于连续目标ν，也可采用显式欧拉格式，但其收敛性保证有限。