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## Regularity of the Monge-Ampère equation in Besov's space

Calculus of Variations and Partial Differential Equations. 2014. Vol. 49. No. 3-4. P. 1187-1197.

Kolesnikov A., Tikhonov S. Y.

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 \Phi \cdot e^{-W(\nabla \Phi)} in the Besov spaces W^{\gamma,1}_{loc}. We prove that D^2 \Phi \in W^{\gamma,1}_{loc} provided e^{-V} belongs to a proper Besov class and W is convex. In particular, D^2 \Phi \in L^p_{loc} for some p>1. Our proof does not rely on the previously known regularity results.

Kolesnikov A., Klartag B., / Cornell University. 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., 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

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

Kolesnikov A., / 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

Caglar U., Kolesnikov A., Werner E., Indiana University Mathematics Journal 2022 Vol. 71 No. 6 P. 2309-2333

In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities ...

Added: June 23, 2023

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

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., / 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

Kolesnikov A., Werner E., Caglar U., / Cornell University. Series arXiv "math". 2021.

We further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities, new affine invariant entropy inequalities and new inequalities on functional affine surface area The functional inequalities lead to new affine invariant inequalities for convex bodies. Equality characterizations in these inequalities are related to a Monge Amp`ere differential ...

Added: December 4, 2021

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

Klartag B., Kolesnikov A., / 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

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

Smirnov E., Пенков И., Игнатьев М. В. et al., М. : ВИНИТИ РАН, 2018

Сборник трудов семинара по алгебре и геометрии Самарского государственного университета ...

Added: August 19, 2018

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

Carlier G., Eichinger K., Kroshnin A., SIAM Journal on Mathematical Analysis 2021 Vol. 53 No. 5 P. 5880-5914

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [J. Bigot, E. Cazelles, and N. Papadakis, SIAM J. Math. Anal., 51 (2019), pp. 2261--2285] as a regularization of Wasserstein barycenters [M. Agueh and G. Carlier, SIAM J. Math. Anal., 43 (2011), pp. 904--924]. After characterizing these barycenters in terms of a system of Monge--Ampère ...

Added: October 27, 2021

Artamonov S., Руновский К., Шмайссер Х., Математические заметки 2020 Т. 108 № 4 С. 617-621

Added: November 4, 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

Kolesnikov A., Богачев В. И., 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

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

Мейрманов А. М., Гальцев О. В., Гальцева О. А., Сибирский математический журнал 2019 Т. 60 № 2 С. 419-428

We consider the problem with free (unknown) boundary for the one-dimensional diffusion-convection equation. The unknown boundary is found from the additional condition on the free boundary. A dilation of the variables reduces the problem to an initial-boundary value problem for a strictly parabolic equation with unknown coefficients in the known domain. These coefficients are found ...

Added: October 30, 2020

Agranovich M. S., Functional Analysis and Its Applications 2011 Vol. 45 No. 1 P. 1-12

We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces Hs , the simplest L2-spaces of the Sobolev type, with the use ...

Added: April 12, 2012

Kolesnikov A., Теория вероятностей и ее применения 2012 Т. 57 № 2 С. 296-321

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 $\R^d$.
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 of $\mu$. In ...

Added: February 19, 2013

Chamorro D., Menozzi S., Potential analysis 2018 Vol. 49 No. 1 P. 1-35

Within the global setting of singular drifts in Morrey-Campanato spaces presented
in Chamorro and Menozzi (Revista Matem´atica Iberoamericana 32(N◦4): 1445–1499
2016) we study now the H¨older regularity properties of the solutions of a transport-diffusion
equation with nonlinear singular drifts that satisfy a Besov stability property. We will see
how this Besov information is relevant and how it allows to ...

Added: December 3, 2018

Kosov E., Fractional Calculus and Applied Analysis 2019 Vol. 22 No. 5 P. 1249-1268

We study fractional smoothness of measures on R^k, that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii--Besov fractional regularity of these distributions under some weak nondegeneracy assumption. ...

Added: December 27, 2019

Popova S., Функциональный анализ и его приложения 2023

Рассматривается задача Канторовича оптимальной транспортировки мер в случае, когда функция стоимости и маргинальные распределения непрерывно зависят от параметра со значениями в метрическом пространстве. Доказывается существование приближенных оптимальных отображений Монжа, непрерывных по параметру. ...

Added: September 13, 2023