Weak topology and Opial property in Wasserstein spaces, with applications to gradient flows and proximal point algorithms of geodesically convex functionals
Emanuele Naldi
TU Braunschweig, GermanyGiuseppe Savaré
Bocconi University, Milano, Italy
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Abstract
In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space of Borel probability measures with finite quadratic moment on a separable Hilbert space .
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 and of minimizers of a lower semicontinuous and geodesically convex functional attaining its minimum. In particular, we will show that every solution to the Wasserstein gradient flow of weakly converge to a minimizer of as the time goes to . Similarly, if is also convex along generalized geodesics, every sequence generated by the proximal point algorithm converges to a minimizer of with respect to the weak topology of .
Cite this article
Emanuele Naldi, Giuseppe Savaré, Weak topology and Opial property in Wasserstein spaces, with applications to gradient flows and proximal point algorithms of geodesically convex functionals. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 4, pp. 725–750
DOI 10.4171/RLM/955