The chaotic order A B among positive invertible operators A,B > 0 on a Hilbert space is introduced by logA≥ logB. Using Uchiyama’s method and Furuta’s Kantorovich-type inequality, we will point out that A B if and only if ‖BpA−p/2B−p/2‖Ap ≥ Bp holds for any 0 < p < p0, where p0 is any fixed positive number. On the other hand, for any fixed p0 > 0, we also show that there exist positive invertib...

By a classical result of Gray et al. (1975) the %̄ distance between stationary processes is identified with an optimal stationary coupling problem of the corresponding stationary measures on the infinite product spaces. This is a modification of the optimal coupling problem from Monge–Kantorovich theory. In this paper we derive some general classes of examples of optimal stationary couplings whi...

The many-body Coulomb repulsive energy of strictly correlated electrons provides direct information on the exact Hohenberg-Kohn exchange-correlation functional in the strong interaction limit. Until now the treatment of strictly correlated electrons has been based on the calculation of comotion functions with the help of semianalytic formulations. This procedure is system-specific and has been ...

The original transport problem is to optimally move a pile of soil to an excavation. Mathematically, given two measures of equal mass, we look for an optimal map that takes one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed this problem by considering a measure whose marginals agree with given two measures instead of a bijection. This generalization lin...

Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Abstract We obtain solutions of the nonlinear degenerate parabolic equation ∂ ρ ∂ t = div ρ ∇c ⋆ [ ∇ (F ′ (ρ) + V) ] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires less uniform convexity assumption than that imposed by Alt and ...

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators $L^2(\mathbf{R}^d)$, and used to estimate convergence rate various asymptotic theories context quantum mechanics. The present work proves a Kantorovich type ...

Some new matrix versions of Kantorovich-Type inequalities for Hermitian matrix are proposed in this paper. We consider what happens to these inequalities when the positive definite matrix is allowed to be positive semidefinite singular or indefinite.