Weak topology and Opial property in Wasserstein spaces, with applications to gradient flows and proximal point algorithms of geodesically convex functionals

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In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space (\mathcal{P}_2(\mathsf{H}),W_2) of Borel probability measures with finite quadratic moment on a separable Hilbert space \mathsf{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 characterising 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(\mathsf{H}) and of minimizers of a lower semicontinuous and geodesically convex functional \phi:\mathcal{P}_2(\mathsf{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(\mathsf{H}) .