A Geometric Study of Wasserstein Spaces: An Addendum on the Boundary

Abstract

We extend the geometric study of the Wasserstein space W2(X) of a simply connected, negatively curved metric space X by investigating which pairs of boundary points can be linked by a geodesic, when X is a tree.Let X be a Hadamard space, by which we mean that X is a complete globally CAT(0), locally compact metric space. Mainly, X is a space where triangles are "thin": points on the opposite side to a vertex are closer to the vertex than they would be in the Euclidean plane. This assumption can also be interpreted as X having non-positive curvature, in a setting more general than manifolds; it has a lot of consequences (the distance is convex, X is contractible, it admits a natural boundary and an associated compactification, . . . ) An important example of Hadamard space, on which we shall focus in this paper, is simply an infinite tree.The set of Borel probability measures of X having finite second moment can be endowed with a natural distance defined using optimal transportation, giving birth to the Wasserstein space W 2 (X). It is well-known that W 2 (X) does not have non-positive curvature even when X is a tree.This note is an addendum to [BK12], where we defined and studied the boundary of W 2 (X). We refer to that article and references therein for the background both on Hadamard space and optimal transportation, as well as for notations. Note that a previous (long) version of [BK12] contained the present content, but has been split after remarks of a referee.Let us quickly sum up the content of [BK12]. The boundary of X can be defined by looking at geodesic rays, and identifying rays that stay at bounded distance one to another ("asymptote" relation). We showed that there is a natural boundary ∂ W 2 (X) of the Wasserstein space that is both close to the traditional boundary of Hadamard spaces (a boundary point can be defined as an asymptote class of rays) and relevant to optimal transportation (a boundary point can be seen as a measure on the cone over ∂X, encoding the asymptotic direction and speed distribution of the mass along a ray). This boundary can

Energy Flow in Particle Collisions

“…More speculatively, it would be interesting to circumvent the (unphysical) particle-level stage of calculations and make theoretical predictions directly in the space of events. Understanding and expanding in this direction would require natural notions of volume and integration in this space, perhaps aided by recent developments in Wasserstein spaces [184][185][186][187], though we leave this fascinating exploration to future work. Nonetheless, it has already been established that the energy flows themselves obey factorization theorems in effective field theory contexts [26], and give rise to rich behavior in correlators [16,37,188,189].…”