In optimization and variational analysis, the proximity operator plays a fundamental role. Within the setting of the Wasserstein space, where it is known as the JKO operator, its non-expansivity is a central topic of investigation, naturally motivated by the classical behavior of convex functionals in Hilbert spaces.

While non-expansivity holds for a class of functionals which satisfy a very strong notion of convexity, the situation is not totally understood for weaker notions of convexity. In this talk, we will explore different stability results under various notions of convexity available in the Wasserstein space.

It is on a joint work with S. Di Marino and E. Naldi.