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## The multistochastic Monge–Kantorovich problem

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 solution.

Gladkov N., Kolesnikov A., Zimin A., On multistochastic Monge-Kantorovich problem, bitwise operations, and fractals / Cornell University. Series arXiv "math". 2018.

The multistochastic (n,k)-Monge--Kantorovich problem on a product space ∏ni=1Xi is an extension of the classical Monge--Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×…×Xik for all k-tuples {i1,…,ik}⊂{1,…,n} for a given 1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution π to the following important model case: n=3,k=2,Xi=[0,1], ...

Added: July 31, 2018

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., 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

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

Kolesnikov A., Elisabeth Werner, Advances in Mathematics 2022 Vol. 396 Article 108110

Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke–Santaló inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a “pointwise Prékopa–Leindler inequality” and show a monotonicity property of the multimarginal Blaschke–Santaó functional. ...

Added: December 4, 2021

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

Gladkov N., Kolesnikov A., Zimin A., Calculus of Variations and Partial Differential Equations 2019 Vol. 58 No. 173 P. 1-33

The multistochastic (n, k)-Monge–Kantorovich problem on a product space ∏ni=1Xi∏i=1nXi is an extension of the classical Monge–Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×⋯×XikXi1×⋯×Xik for all k-tuples {i1,…,ik}⊂{1,…,n}{i1,…,ik}⊂{1,…,n} for a given 1≤k<n1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution ππ to the following important model ...

Added: October 9, 2019

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

Springer Nature Switzerland AG, 2019

Gathering the proceedings of the 11th CHAOS2018 International Conference, this book highlights recent developments in nonlinear, dynamical and complex systems. The conference was intended to provide an essential forum for Scientists and Engineers to exchange ideas, methods, and techniques in the field of Nonlinear Dynamics, Chaos, Fractals and their applications in General Science and the ...

Added: October 29, 2021

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

Kolesnikov A., Владимир Игоревич Богачев, Доклады Академии наук 2012 Т. 44 № 2 С. 131-136

Работа связана с изучением соболевской регулярности отображений
оптимальной транспортировки в бесконечномерных пространствах, наделенных гауссовской мерой. Найдены условия принадлежности соболевскому классу для таких отображений. Доказана формула замены переменных. ...

Added: February 19, 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

Linetskiy B., Sedova T.L., Tikhonov А. N. et al., , in: Innovative Information Technologies: Materials of the International scientific–practical conference. Part 2. * 2.: M.: HSE, 2014.. P. 219-226.

This article consider The project of the scientific and educational Center for integration of multimedia technologies in science, education and culture, as space-technological environment for the implementation of innovative scientific and educational projects of the 21st century, which should become the support for the master's programs, especially interdisciplinary; at the intersection of science, art and ...

Added: May 21, 2014

Kolesnikov A., Zimin A., Sandomirskiy F. et al., Beckmann's approach to multi-item multi-bidder auctions / Cornell University. Series Theoretical Economics "arxiv.org". 2022. No. 2203.06837.

We consider the problem of revenue-maximizing Bayesian auction design with several i.i.d. bidders and several items. We show that the auction-design problem can be reduced to the problem of continuous optimal transportation introduced by Beckmann. We establish the strong duality between the two problems and demonstrate the existence of solutions. We then develop a new ...

Added: April 10, 2022

Gladkov N., Kolesnikov A., Zimin A., The multistochastic Monge-Kantorovich problem / Cornell University. Series arXiv "math". 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 ...

Added: August 21, 2020

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

Molchanov S., Joe Chen, Alexander Teplyaev, Journal of Physics A: Mathematical and Theoretical 2015 Vol. 48 No. 39 P. 1-22

...

Added: June 22, 2016

Bogachev V., Александр Николаевич Калинин, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 463 № 4 С. 383-386

Установлены точные условия равенства минимумов в задачах Монжа и Канторовича ...

Added: November 15, 2017

Trubochkina N. K., Анастасия Владимировна Лиховцева, Мир техники кино / World of Technique of Cinema (WTC). Россия. ISSN: 1991-3400 2015 № 2(36) С. 11-18

The article describes a method of producing three-dimensional video using fractal mathematics and information technologies. Various 3D visualization technologies of three-dimensional fractals, both in static and dynamic form are considered. Methods of three-dimensional fractal video synthesis used for various applications are described. One of the applications is a 3D cinema with glasses, where the background ...

Added: July 10, 2015

Silaev A. M., International Research Journal of Finance and Economics 2012 No. 100 P. 12-18

The paper introduces the idea of using fractals for stylized facts modelling. Since the behavior of financial markets is hard to predict, there have appeared new financial assets models, based, first of all, on empirically revealed properties or stylized empirical facts. We have found the correspondence between four types of fractals and four main empirical ...

Added: November 6, 2012

Decrouez G. G., Pierre Olivier Amblard, Jean Marc Brossier et al., Journal of Physics A: Mathematical and Theoretical 2009 Vol. 42 No. 9 P. 1-17

Iterated function systems (IFS) are interesting parametric models for generating fractal sets and functions. The general idea is to compress, deform and translate a given set or function with a collection of operators and to iterate the procedure. Under weak assumptions, IFS possess a unique fixed point which is in general fractal. IFS were introduced ...

Added: October 2, 2014

Bogachev V., Колесников А., Успехи математических наук 2012 Т. 67 № 5 С. 3-110

Дан обзор совеременного состояния исследований, связанных с задачами Монжа и Канторовича оптимальной транспортировки мер. ...

Added: February 26, 2014

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