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Wasserstein Asymptotics for Brownian Motion on the Flat Torus and Brownian Interlacements
Stochastic Processes and their Applications. 2025. Vol. 183.
Mariani M., Trevisan D.
Keywords: Optimal transportation
Kolesnikov A., / Series arXiv "math". 2023.
In this survey paper we present classical and recent results relating the auction design and the optimal transportation theory. ...
Added: December 13, 2023
Gladkov N., Kolesnikov A., Zimin A., / 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
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
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
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
Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem
Kolesnikov A., / 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., 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
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
Kosorukov O., Journal of Computer and Systems Sciences International 2016 Vol. 55 No. 6 P. 1010–1015
In this paper formulations of the synthesis and analysis problems in multiterminal comunication networks with variable edge capacities are considered. ...
Added: August 17, 2017
Klartag B., Kolesnikov A., Analysis Mathematica 2017 Vol. 43 No. 1 P. 67–88
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 Kähler–Einstein equation eΦ = detD2Φ on proper convex cones. We prove a generalization of this theorem by showing ...
Added: April 17, 2017
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
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 <img />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 ...
Added: December 23, 2015
Kolesnikov A., Milman E., 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., Tikhonov S. Y., Calculus of Variations and Partial Differential Equations 2014 Vol. 49 No. 3-4 P. 1187–1197
Let \mu = e^{-V} \ dx be a probability measure and T = \nabla \Phi be the optimal transportation mapping pushing forward \mu onto a log-concave compactly supported measure \nu = e^{-W} \ dx. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{\`e}re equation e^{-V} = \det D^2 ...
Added: March 12, 2014
Kolesnikov A., Klartag B., / Series math "arxiv.org". 2013. No. 1402.2636.
We investigate the Brenier map \nabla \Phi between the uniform measures on two convex domains in \mathbb{R}^n or more generally, between two log-concave probability measures on \mathbb{R}^n. We show that the eigenvalues of the Hessian matrix D^2 \Phi exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or ...
Added: March 12, 2014
Kolesnikov A., Discrete and Continuous Dynamical Systems 2014 Vol. 34 No. 4 P. 1511–1532
We study the optimal transportation mapping VΦ: ℝd → ℝd pushing forward a probability measure μ = e -V dx onto another probability measure ν = e-W dx. Following a classical approach of E. Calabi we introduce the Riemannian metric g = D2 Φ on ℝd and study spectral properties of the metric-measure space M ...
Added: November 12, 2013
Kudryavtseva O., Nagapetyan T., Kolesnikov A., Journal Mathematical Economics, Netherlands 2013 Vol. 49 P. 501–505
The famous Afriat’s theorem from the theory of revealed preferences establishes necessary and sufficient conditions for the existence of utility function for a given set of choices and prices. The result on the existence of a homogeneous utility function can be considered as a particular fact of the Monge–Kantorovich mass transportation theory. In this paper ...
Added: September 27, 2013
Kolesnikov A., Zaev D., / 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
Kolesnikov A., / 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
Kolesnikov A., / 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., Bogachev V. I., 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