On the stability of proximal operators in Wasserstein spaces under different notions of convexity

The proximal operator is a fundamental tool in variational analysis and optimization.

In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous.

In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and Adve and Mészáros, among others, and are not completely understood.

In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.