Orateur
Description
It has been known since Otto’s seminal work that the space of
probability measures $\mathcal{P}(M)$ on a closed Riemannian manifold $M$ can be
endowed with an infinite-dimensional, quasi-Riemannian structure. After briefly
recalling some fundamental notions from Riemannian geometry, we will explore the
resulting geometry of $\mathcal{P}(M)$. Despite its formal Riemannian structure, the
geometry of $\mathcal{P}(M)$ is highly singular. To overcome these singularities, we
will focus on the subspace $\mathcal{P}_\infty(M)$ of smooth, positive probability
measures.
A central and notoriously difficult question is whether $\mathcal{P}_\infty(M)$ is
geodesically convex. We will explain how entropic interpolations provide a natural
and effective approximation of Wassertein geodesics within $\mathcal{P}_\infty(M)$.
If time permits, we will conclude by discussing an application of this approximation
result to a curvature condition on $\mathcal{P}_\infty(M)$, with consequences for
the stability of $\mathcal{P}_\infty$-valued Stochastic Differential Equations.