We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_ν:= \text{MMD}_K^2(\cdot, ν)$ towards given target measures $ν$ 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 $\mathcal F_ν$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $ν$, 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 $\mathcal F_ν$-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 $ν$, also the explicit Euler scheme can be employed, although with limited convergence guarantees.

翻译：本文系统描述了实直线上最大平均差异（MMD）泛函$\mathcal F_ν:= \text{MMD}_K^2(\cdot, ν)$向给定目标测度$ν$的Wasserstein梯度流，重点研究负距离核$K(x,y) := -|x-y|$。在一维情形下，Wasserstein-2空间可等距嵌入到分位数函数构成的锥$\mathcal C(0,1) \subset L_2(0,1)$中，从而通过求解$L_2(0,1)$上的相关柯西问题来刻画Wasserstein梯度流。基于在$L_2(0,1)$上构造$\mathcal F_ν$的适当对应形式及其次微分，我们给出了该柯西问题的解。对于离散目标测度$ν$，该解可表示为分段线性公式。我们证明了该流在$\mathcal C(0,1)$子集上的不变性与光滑性，对于某些$\mathcal F_ν$-流这意味着初始点测度会瞬时变为绝对连续并持续保持。最后，采用易于通过二分法计算的隐式欧拉格式，通过多种数值算例展示了该流的动态行为。对于连续目标$ν$，虽然收敛性保证受限，仍可使用显式欧拉格式。