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 $(\mathcal{P}_2(\mathsf{H}),W_2)$ of Borel probability measures with finite quadratic moment on a separable Hilbert space $\mathsf{H}$.