Second order analysis on \((\mathcal P_{2}(M),W_{2})\) (Q2880225)

In the last 15 years, many papers have been devoted, starting from the seminal contributions of Otto and Benamou-Brenier, to the description of the space of probability measures on a Riemannian manifold \(M\) as an infinite-dimensional Riemannian manifold. Using the continuity equation and the metric NEWLINE\[NEWLINE g_\mu(v,w):=\int_M \langle v,w\rangle\,d\mu NEWLINE\]NEWLINE on the velocity fields, which are thought of as tangent vectors, it turns out that the induced Riemannian distance is precisely the optimal transportation distance with a quadratic cost function. A first-order description (geodesics, characterization of absolutely continuous curves and of the tangent vector) has been completely achieved in the monograph by Ambrosio, Gigli and Savaré in the Euclidean space, and later extended to Riemannian manifolds. The aim of this monograph is to extend the analysis to second-order, namely to discussions of the notions of curvature, parallel transport and Jacobi fields. A first contribution in this direction is \textit{J. Lott}'s paper [Commun. Math. Phys. 277, No. 2, 423--437 (2008; Zbl 1144.58007)], where this analysis is performed at a formal level, but with a far reaching description of the full curvature tensor. A second one is [[1]: \textit{L. Ambrosio} and \textit{N. Gigli}, Methods Appl. Anal. 15, No. 1, 1--29 (2008; Zbl 1179.28009)], where a rigorous construction of the parallel transport is given in the case \(M={\mathbb R}^n\). The results of this paper extend those of [1] to complete, connected and \(C^\infty\) Riemannian manifolds \(M\) without boundary. The analysis performed here is quite complete: parallel transport, characterization of the Levi-Civita connection and description of the curvature tensor, differentiability of the transport map and existence of the Jacobi fields. Also, the regularity issues related to the potential lack of regularity of the optimal maps are clearly illustrated.