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## The multistochastic Monge-Kantorovich problem

math.
arXiv.
Cornell University
,
2020.

The multistsochastic Monge--Kantorovich problem on the product $X = \prod_{i=1}^n X_i$ of $n$ spaces is a generalization of the multimarginal Monge--Kantorovich problem. For a given integer number $1 \le k<n$ we consider the minimization problem $\int c d \pi \to \inf$ of the space of measures with fixed projections onto every $X_{i_1} \times \dots \times X_{i_k}$ for arbitrary set of $k$ indices $\{i_1, \dots, i_k\} \subset \{1, \dots, n\}$. In this paper we study basic properties of the multistochastic problem, including well-posedness, existence of a dual solution, boundedness and continuity of a dual solution.

Publication based on the results of:

Gladkov N., Zimin A., SIAM Journal on Mathematical Analysis 2020 Vol. 52 No. 4 P. 3666-3696

We construct an explicit solution for the multimarginal transportation problem on the unit cube $[0, 1]^3$ with the cost function $xyz$ and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with a nonconstant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique ...

Added: August 21, 2020

Kolesnikov A., Weak regularity of Gauss mass transport / Cornell University. Series math "arxiv.org". 2009. No. 0904.1852.

Given two probability measures $\mu$ and $\nu$ we consider a mass transportation mapping $T$ satisfying 1) $T$ sends $\mu$ to $\nu$, 2) $T$ has the form $T = \phi \frac{\nabla \phi}{|\nabla \phi|}$, where $\phi$ is a function with convex sublevel sets.
We prove a change of variables formula for $T$. We also establish Sobolev estimates for ...

Added: March 27, 2013

Kolesnikov A., Zaev D., Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry. / Cornell University. Series math "arxiv.org". 2013.

We consider probability measures on $\mathbb{R}^{\infty}$ and study natural analogs of optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric properties, in particular, exchangeable and stationary measures. It turns out that the existence problem for optimal transportation is closely related to various ergodic ...

Added: May 13, 2013

Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem

Kolesnikov A., Moscow Mathematical Journal 2020 Vol. 20 No. 1 P. 67-91

We study the transportation problem on the unit sphere Sn−1 for symmetric probability measures and the cost function c(x,y)=log1⟨x,y⟩.
We calculate the variation of the corresponding Kantorovich functional K and study a naturally associated metric-measure space on Sn−1 endowed with a Riemannian
metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are
solutions to the symmetric log-Minkowski problem and ...

Added: October 9, 2019

Kolesnikov A., Theory of Probability and Its Applications 2013 Vol. 57 No. 2 P. 243-264

We study Sobolev a priori estimates for the optimal transportation $T = \nabla \Phi$ between probability measures $\mu=e^{-V} \, dx$ and $\nu=e^{-W} \, dx$ on ${\bf R}^d$. Assuming uniform convexity of the potential $W$ we show that $\int \| D^2 \Phi\|^2_{HS} \, d\mu$, where $\|\cdot\|_{HS}$ is the Hilbert--Schmidt norm, is controlled by the Fisher information ...

Added: December 23, 2015

Kolesnikov A., Lysenko N. Y., Remarks on mass transportation minimizing expectation of a minimum of affine functions / Cornell University. Series arXiv "math". 2015.

We study Monge-Kantorovich problem with one-dimensional marginals μ,ν and the cost function c=min{l1,…,ln} which equals to minimum of a finite number n of affine functions li satisfying certain non-degeneracy assumptions. We prove that the problem is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated on the union of n products Ii×Ji, where {Ii}, {Ji} are partitions of the line into unions ...

Added: February 23, 2016

Nikita Gladkov, Kolesnikov A., Zimin A., Journal of Mathematical Analysis and Applications 2022 Vol. 506 No. 2 Article 125666

The multistochastic Monge–Kantorovich problem on the product X=∏i=1nXi of n spaces is a generalization of the multimarginal Monge–Kantorovich problem. For a given integer number 1≤k<n we consider the minimization problem ∫cdπ→inf on the space of measures with fixed projections onto every Xi1×…×Xik for arbitrary set of k indices {i1,…,ik}⊂{1,…,n}. In this paper we study basic properties of the multistochastic problem, including well-posedness, existence of a dual solution, boundedness and continuity of a dual ...

Added: December 4, 2021

Kolesnikov A., Zaev D., Theory of Stochastic Processes 2015 Vol. 20(36) No. 2 P. 54-62

We study the Monge and Kantorovich transportation problems on R∞R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on a Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, ...

Added: July 8, 2016

Kolesnikov A., Lysenko N. Y., Theory of Stochastic Processes 2016 Vol. 21(37) No. 2 P. 22-28

We study the Monge--Kantorovich problem with one-dimensional marginals $\mu$ and $\nu$ and
the cost function $c = \min\{l_1, \ldots, l_n\}$
that equals the minimum of a finite number $n$ of affine functions $l_i$
satisfying certain non-degeneracy assumptions. We prove that the problem
is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated
on ...

Added: December 30, 2017

Kolesnikov A., Hessian manifolds, CD(K,N) -spaces, and optimal transportation of log-concave measures / Cornell University. Series math "arxiv.org". 2012. No. 1201.2342.

We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the ...

Added: March 28, 2013

Колесников А., Bulletin des Sciences Mathematiques 2014 Vol. 138 No. 2 P. 165-198

Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν, 2) T has the form T=φ∇φ|∇φ|, where φ is a function with convex sublevel sets. We prove a change of variables formula for T. We also establish Sobolev estimates for φ, and a ...

Added: December 23, 2015

Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem

Kolesnikov A., Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem / Cornell University. Series arXiv "math". 2018.

We study the transportation problem on the unit sphere Sn−1 for symmetric probability measures and the cost function c(x,y)=log1⟨x,y⟩. We calculate the variation of the corresponding Kantorovich functional K and study a naturally associated metric-measure space on Sn−1 endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric ...

Added: July 31, 2018

Kolesnikov A., Zaev D., Exchangeable optimal transportation and log-concavity / Cornell University. Series arXiv "math". 2015.

We study the Monge and Kantorovich transportation problems on R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, ...

Added: February 23, 2016

Kolesnikov A., Bulletin des Sciences Mathematiques 2014 Vol. 138 No. 2 P. 165-198

Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν , 2) T has the form T=ϕ∇ϕ|∇ϕ| , where ϕ is a function with convex sublevel sets. We prove a change of variables formula for T . We also establish Sobolev estimates for ϕ ...

Added: February 24, 2016

Lysenko N. Y., Mathematical notes 2014

It is known that the Az\'ema-Yor solution to the Skorokhod embedding problem maximizes the law of the running maximum of an uniformly integrable martingale with given terminal value distribution. Recently this optimality property has been generalized to expectations of certain bivariate functions depending on the terminal value and the running maximum. In the present paper ...

Added: December 12, 2014

Zaev D., Kolesnikov A., Kyoto Journal of Mathematics 2017 Vol. 57 No. 2 P. 293-324

We consider probability measures on R∞ and study optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric properties, in particular, exchangeable and stationary measures. We show in the latter case that existence problem for optimal transportation is closely related to ergodicity of the target measure. ...

Added: December 30, 2017

Zaev D., On the Monge-Kantorovich problem with additional linear constraints / Cornell University. Series math "arxiv.org". 2014.

We consider the modified Monge-Kantorovich problem with additional restriction: admissible transport plans must vanish on some fixed functional subspace. Different choice of the subspace leads to different additional properties optimal plans need to satisfy. Our main results are quite general and include several important examples. In particular, they include Monge-Kantorovich problems in the classes of ...

Added: May 14, 2014

Bo'az Klartag, Kolesnikov A., Remarks on curvature in the transportation metric / Cornell University. Series math "arxiv.org". 2016.

According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the ``hyperbolic" toric K\"ahler-Einstein equation $e^{\Phi} = \det D^2 \Phi$ on proper convex cones. We prove a generalization of this ...

Added: April 14, 2016

Zimin A., Gladkov N., An explicit solution for a multimarginal mass transportation problem / Cornell University. Series arXiv "math". 2018.

We construct an explicit solution for the multimarginal transportation problem on the unit cube [0,1]3 with the cost function xyz and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with non-constant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique (up to ...

Added: October 10, 2018

Kolesnikov A., Emanuel Milman, Geometric Aspects of Functional Analysis, Israel Seminar 2014 Vol. 2116 P. 273-293

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body Ω ⊂ R n , not necessarily vanishing on the boundary ∂Ω. This reduces the study of the Neumann Poincar´e constant on Ω to that of the cone and Lebesgue measures on ∂Ω; these may be bounded via the curvature ...

Added: April 13, 2015

Kolesnikov A., V.I. Bogachev, Russian Mathematical Surveys 2012 Vol. 67 No. 5 P. 785-890

This article gives a survey of recent research related to the Monge-Kantorovich problem. Principle results are presented on the existence of solutions and their properties both in the Monge optimal transportation problem and the Kantorovich optimal plan problem, along with results on the connections between both problems and the cases when they are equivalent. Diverse ...

Added: February 12, 2013

Kolesnikov A., Olga Kudryavtseva, Tigran Nagapetyan, Remarks on the Afriat's theorem and the Monge-Kantorovich problem / Cornell University. Series math "arxiv.org". 2013.

The classical concept of the revealed preferences was introduced by P. Samuelson and studied by H.S. Houthakker, M. Richter, S. Afriat, H. Varian and many others. It was shown by Afriat that the so called SARP (or cyclically consistence) axiom is a necessary and sufficient condition for existence of an appropriate concave utility function for ...

Added: February 23, 2013

191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...

Added: September 23, 2016

Losev A. S., S. Slizovskiy, JETP Letters 2010 Vol. 91 P. 620-624

Added: February 27, 2013