Weak topology and Opial property in Wasserstein spaces, with applications to Gradient Flows and Proximal Point Algorithms of geodesically convex functionals

Abstract: 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 $\phi:\mathcal{P}_2(H)\to(-\infty,+\infty]$ attaining its minimum. In particular, we will show that every solution to the Wasserstein gradient flow of $\phi$ weakly converge to a minimizer of $\phi$ as the time goes to $+\infty$. Similarly, if $\phi$ is also convex along generalized geodesics, every sequence generated by the proximal point algorithm converges to a minimizer of $\phi$ with respect to the weak topology of $\mathcal{P}_2(H)$.