Narrow Results

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations.

A Wasserstein space is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be made precise.

In the first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a natural set of spaces generalizing the usual Hilbert cube. These invariants are very similar to concepts initiated by Rogers, but our variant is specifically suited to tackle Lipschitz comparison.

In the second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide uniform bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d. These arguments are very easily adapted to study the space of closed subsets of a compact metric space, partly generalizing results of Boardman, Goodey and McClure.

The aim of this paper is to obtain a better understanding of the geometric structure of quadratic Wasserstein spaces over separable Hilbert spaces. For this sake, we focus on their cone and product structures, and prove that the quadratic Wasserstein space over any separable Hilbert space has a cone structure and splits the underlying space isometrically but no more than that. These are shown in more general settings, and one of our main results is that the quadratic Wasserstein space over a Polish space has a cone structure if and only if so does the underlying space.

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space .

In this paper we investigate the geometry of when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, is not non-positively curved, our results show that have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in .