Wasserstein空间中的弱拓扑和Opial性质及其在测地线凸泛函的梯度流和邻近点法中的应用
Weak topology and Opial property in Wasserstein spaces, with applications to Gradient Flows and Proximal Point Algorithms of geodesically convex functionals
摘要：
本文讨论了在可分Hilbert空间$H$上的有限二次矩Borel概率测度的Wasserstein空间$(\mathcal{P}_2(H),W_2)$中如何定义一个适当的弱拓扑概念.我们将证明这样的拓扑继承了Hilbert空间中通常的弱拓扑的许多特征，特别是测地凸闭集的弱闭性和弱收敛序列的Opial性质。我们将这一概念应用于$\mathcal{P}_2(H)$的弱闭子集中的非扩张映射的不动点的逼近以及下连续测地凸泛函$ϕ:\mathcal{P}_2(H)\to(-\infty,+\infty]$的极小值的逼近.特别地，我们将证明$ϕ$的Wasserstein梯度流的每一个解随着时间的推移到$+\infty$弱收敛到$ϕ$的极小值。类似地，如果$ϕ$也是沿广义测地线凸的，则由邻近点法生成的每个序列都收敛到$ϕ$关于$\mathcal{P}_2(H)$的弱拓扑的极小元。
英文摘要：
In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space $(\mathcal{P}_2(H),W_2)$ of Borel probability measures with finite quadratic moment on a separable Hilbert space $H$. We will show that such a topology inherits many features of the usual weak topology in Hilbert spaces, in particular the weak closedness of geodesically convex closed sets and the Opial property characterizing weakly convergent sequences. We apply this notion to the approximation of fixed points for a non-expansive map in a weakly closed subset of $\mathcal{P}_2(H)$ and of minimizers of a lower semicontinuous and geodesically convex functional $ϕ:\mathcal{P}_2(H)\to(-\infty,+\infty]$ attaining its minimum. In particular, we will show that every solution to the Wasserstein gradient flow of $ϕ$ weakly converge to a minimizer of $ϕ$ as the time goes to $+\infty$. Similarly, if $ϕ$ is also convex along generalized geodesics, every sequence generated by the proximal point algorithm converges to a minimizer of $ϕ$ with respect to the weak topology of $\mathcal{P}_2(H)$.
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