Brendan Pass

Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem

We study a multi-marginal optimal transportation problem. Under certain conditions on the cost function and the first marginal, we prove that the solution to the relaxed, Kantorovich version of the problem induces a solution to the Monge problem and that the solutions to both problems are unique.

Infinite-body optimal transport with Coulomb cost

We introduce and analyze symmetric infinite-body optimal transport (OT) problems with cost function of pair potential form. We show that for a natural class of such costs, the optimizer is given by the independent product measure all of whose factors are given by the one-body marginal. This is in striking contrast to standard finite-body OT problems, in which the optimizers are typically highly correlated, as well as to infinite-body OT problems with Gangbo–Swiech cost. Moreover, by adapting a construction from the study of exchangeable processes in probability theory, we prove that the corresponding

A General Condition for Monge Solutions in the Multi-Marginal Optimal Transport Problem

N

Remarks on the semi-classical Hohenberg–Kohn functional

N-body OT problem is well approximated by the infinite-body problem. To our class belongs the Coulomb cost which arises in many-electron quantum mechanics. The optimal cost of the Coulombic N-body OT problem as a function of the one-body marginal density is known in the physics and quantum chemistry literature under the name SCE functional, and arises naturally as the semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results imply that in the inhomogeneous high-density limit (i.e.

On a Class of Optimal Transportation Problems with Infinitely Many Marginals

N\rightarrow \infty

Convexity and multi-dimensional screening for spaces with different dimensions

N→∞ with arbitrary fixed inhomogeneity profile