| Abstract: |
| Given a metric space $(X,d)$ and a functional $\mathcal{F}$ defined on it, a relevant role in the theory of gradient flows associated with $\mathcal{F}$ is played by the \emph{proximity operator}
\begin{align*}
P_{\mathcal{F}}(x)\coloneqq \arg\min_{y}\left\{\frac{1}{2}d^2(x,y)+\mathcal{F}(y)\right\}.
\end{align*}
In the setting of the Wasserstein space, that is $(X,d)=(\mathcal{P}_2(\mathbb{R}^d), W_2)$, the proximity operator is often called \emph{JKO operator}.
Motivated from the fact that if $(X,d)$ is a Hilbert space then $x\mapsto P_{\mathcal{F}}(x)$ is non-expansive when $\mathcal{F}$ is convex, the non-expansivity of the $JKO$ operator has been a topic of investigation. It has been proved recently by Cavagnari-Savare`-Sodini that if $\mathcal{F}$ is \emph{totally convex} then non-expansivity holds. Various partial results are present in literature assuming weaker notions of convexity. In this talk we review these contributions and we present some generalizations under the assumption that $\mathcal{F}$ is convex on generalized geodesics. This is based on a joint work in preparation with Di Marino and Naldi. |
|