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 and demonstrate differences to the explicit Euler scheme, which is easier to compute, but comes with limited convergence guarantees.

翻译：我们全面描述了实轴上最大均值差异（MMD）泛函$\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$朝向给定目标测度$\nu$的Wasserstein梯度流，其中聚焦于负距离核$K(x,y) := -|x-y|$。在一维情形下，Wasserstein-2空间可等距嵌入分位数函数锥$\mathcal C(0,1) \subset L_2(0,1)$，从而通过$L_2(0,1)$上关联Cauchy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Cauchy问题的解。对于离散目标测度$\nu$，该解具有分段线性的显式表达式。我们证明了该流在$\mathcal C(0,1)$子集上的不变性与平滑性质。对于特定$\mathcal F_\nu$-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，我们通过隐式Euler格式的数值算例展示了该流的行为特征，并与显式Euler格式进行对比——后者虽更易计算，但收敛性保证有限。