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Alexander V. Kolesnikov

Publications and source records attributed to Alexander V. Kolesnikov.

At least 37 records · Page 2Linked to original sources
arXiv2017

Brascamp-Lieb type inequalities on weighted Riemannian manifolds with boundary

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp-Lieb type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell's convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.

math.DG↗
arXiv2017

Poincaré and Brunn--Minkowski inequalities on the boundary of weighted Riemannian manifolds

We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, originally derived as an infinitesimal form of the Brunn-Minkowski inequality, thereby precluding any extensions beyond the Euclidean setting. A dual version for generalized mean-convex boundaries is also obtained, yielding spectral-gap estimates for the weighted Laplacian on the boundary. Motivated by these inequalities, a new geometric evolution equation is proposed, which extends to the Riemannian setting the Minkowski addition operation of convex domains, a notion thus far confined to the purely linear setting. This geometric flow is characterized by having parallel normals (of varying velocity) to the evolving hypersurface along the trajectory, and is intimately related to a homogeneous Monge-Ampère equation on the exterior of the convex domain. Using the aforementioned Poincaré-type inequality on the boundary of the evolving hypersurface, we obtain a novel Brunn--Minkowski inequality in the weighted-Riemannian setting, amounting to a certain concavity property for the weighted-volume of the evolving enclosed domain. All of these results appear to be new even in the classical non-weighted Riemannian setting.

math.DG↗
arXiv2017

Extremal Kaehler-Einstein metric for two-dimensional convex bodies

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 Φ$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.

math.DG↗
arXiv2017

Remarks on mass transportation minimizing expectation of a minimum of affine functions

We study the Monge--Kantorovich problem with one-dimensional marginals $μ$ and $ν$ 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 the union of $n$ products $I_i \times J_i$, where $\{I_i\}$ and $\{J_i\}$ are partitions of the real line into unions of disjoint connected sets. The families of sets $\{I_i\}$ and $\{J_i\}$ have the following properties: 1) $c=l_i$ on $I_i \times J_i$, 2) $\{I_i\}, \{J_i\}$ is a couple of partitions solving an auxiliary $n$-dimensional extremal problem. The result is partially generalized to the case of more than two marginals.

math.PR↗
arXiv2016

Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities

Given a probability measure $μ$ supported on a convex subset $Ω$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincaré and log-Sobolev type inequalities on $(Ω,g_0,μ)$. To this end, we change the metric $g_0$ to a more general Riemannian one $g$, adapted in a certain sense to $μ$, and perform our analysis on $(Ω,g,μ)$. The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when $μ$ is unconditional, i.e. invariant under reflection with respect to the principle hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking on $(Ω,g,μ)$ tools such as Riemannian generalizations of the Brascamp--Lieb inequality and the Bakry--Émery criterion, and passing back to the original Euclidean metric, we obtain various weighted inequalities on $(Ω,g_0,μ)$: refined and entropic versions of the Brascamp--Lieb inequality, weighted Poincaré and log-Sobolev inequalities, Hardy-type inequalities, etc. Key to our analysis is the positivity of the associated Lichnerowicz--Bakry--Émery generalized Ricci curvature tensor, and the convexity of the manifold $(Ω,g,μ)$. In some cases, we can only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.

math.FA↗
arXiv2015

Exchangeable optimal transportation and log-concavity

We study the Monge and Kantorovich transportation problems on $\mathbb{R}^{\infty}$ 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, under certain analytical assumptions involving log-concavity of the target measure. As a by-product we obtain the following result: any uniformly log-concave exchangeable sequence of random variables is i.i.d.

math.PR↗
arXiv2015

Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry

We consider probability measures on $\mathbb{R}^{\infty}$ 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. In particular, we prove existence of the symmetric optimal transportation for a certain class of stationary Gibbs measures.

math.FA↗
arXiv2015

Remarks on the KLS conjecture and Hardy-type inequalities

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $Ω\subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial Ω$. This reduces the study of the Neumann Poincaré constant on $Ω$ to that of the cone and Lebesgue measures on $\partial Ω$; these may be bounded via the curvature of $\partial Ω$. A second reduction is obtained to the class of harmonic functions on $Ω$. We also study the relation between the Poincaré constant of a log-concave measure $μ$ and its associated K. Ball body $K_μ$. In particular, we obtain a simple proof of a conjecture of Kannan--Lovász--Simonovits for unit-balls of $\ell^n_p$, originally due to Sodin and Latała--Wojtaszczyk.

math.SP↗
arXiv2014

Eigenvalue distribution of optimal transportation

We investigate the Brenier map $\nabla Φ$ 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 $D^2 Φ$ exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension $n$.

math.AP↗
arXiv2013

On continuity equations in infinite dimensions with non-Gaussian reference measure

Let $γ$ be a Gaussian measure on a locally convex space and $H$ be the corresponding Cameron-Martin space. It has been recently shown by L. Ambrosio and A. Figalli that the linear first-order PDE $$ \dotρ + \mbox{div}_γ (ρ\cdot {b}) =0, \ \ ρ|_{t=0} = ρ_0, $$ where $ρ_0 \cdot γ$ is a probability measure, admits a weak solution, in particular, under the following assumptions: $$ \|b\|_{H} \in L^p(γ), \ p>1, \ \ \ \exp\bigl(\varepsilon(\mbox{\rm div}_γ b)_{-} \bigr) \in L^1(γ). $$ Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures $ν$ on $\R^{\infty}$, under the main assumption that $β_i \in \cap_{n \in \Nat} L^{n}(ν)$ for every $i \in \Nat$, where $β_i$ is the logarithmic derivative of $ν$ along the coordinate $x_i$. We also show uniqueness of the solution for a wide class of measures. This class includes uniformly log-concave Gibbs measures and certain product measures. measures.

math.FA↗
arXiv2013

Weak regularity of Gauss mass transport

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 $T = ϕ\frac{\nabla ϕ}{|\nabla ϕ|}$, 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 new form of the parabolic maximum principle. In addition, we discuss relations to the Monge-Kantorovich problem, curvature flows theory, and parabolic nonlinear PDE's.

math.FA↗
arXiv2013

Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures

We study the optimal transportation mapping $\nabla Φ: \mathbb{R}^d \mapsto \mathbb{R}^d$ 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 $g = D^2 Φ$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\mathbb{R}^d, g, μ)$. We prove, in particular, that $M$ admits a non-negative Bakry--{É}mery tensor provided both $V$ and $W$ are convex. If the target measure $ν$ is the Lebesgue measure on a convex set $Ω$ and $μ$ is log-concave we prove that $M$ is a $CD(K,N)$ space. Applications of these results include some global dimension-free a priori estimates of $\| D^2 Φ\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.

math.AP↗
arXiv2013

Remarks on the Afriat's theorem and the Monge-Kantorovich problem

The famous Afriat's theorem from the theory of revealed preferences establishes necessary and suffient conditions for existence of utility function for a given set of choices and prices. The result on existence of a {\it homogeneous} utility function can be considered as a particular fact of the Monge-Kantorovich mass transportation theory. In this paper we explain this viewpoint and discuss some related questions.

math.OC↗
arXiv2013

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

Let $μ= e^{-V} \ dx$ be a probability measure and $T = \nabla Φ$ be the optimal transportation mapping pushing forward $μ$ onto a log-concave compactly supported measure $ν= e^{-W} \ dx$. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{è}re equation $e^{-V} = \det D^2 Φ\cdot e^{-W(\nabla Φ)}$ in the Besov spaces $W^{γ,1}_{loc}$. We prove that $D^2 Φ\in W^{γ,1}_{loc}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2 Φ\in L^p_{loc}$ for some $p>1$. Our proof does not rely on the previously known regularity results.

math.AP↗
arXiv2012

Sobolev regularity for the Monge-Ampere equation in the Wiener space

Given the standard Gaussian measure $γ$ on the countable product of lines $\mathbb{R}^{\infty}$ and a probability measure $g \cdot γ$ absolutely continuous with respect to $γ$, we consider the optimal transportation $T(x) = x + \nabla φ(x)$ of $g \cdot γ$ to $γ$. Assume that the function $|\nabla g|^2/g$ is $γ$-integrable. We prove that the function $φ$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {\det}_2(I + D^2 φ) \exp \bigl(\mathcal{L} φ- 1/2 |\nabla φ|^2 \bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $φ$.

math.FA↗
arXiv2011

On isoperimetric sets of radially symmetric measures

We study the isoperimetric problem for the radially symmetric measures. Applying the spherical symmetrization procedure and variational arguments we reduce this problem to a one-dimensional ODE of the second order. Solving numerically this ODE we get an empirical description of isoperimetric regions of the planar radially symmetric exponential power laws. We also prove some isoperimetric inequalities for the log-convex measures. We show, in particular, that the symmetric balls of large size are isoperimetric sets for strictly log-convex and radially symmetric measures. In addition, we establish some comparison results for general log-convex measures.

math.PR↗
arXiv2011

On Sobolev regularity of mass transport and transportation inequalities

We study Sobolev a priori estimates for the optimal transportation $T = \nabla Φ$ between probability measures $μ=e^{-V} \ dx$ and $ν=e^{-W} \ dx$ on $\R^d$. Assuming uniform convexity of the potential $W$ we show that $\int \| D^2 Φ\|^2_{HS} \ dμ$, where $\|\cdot\|_{HS}$ is the Hilbert-Schmidt norm, is controlled by the Fisher information of $μ$. In addition, we prove similar estimate for the $L^p(μ)$-norms of $\|D^2 Φ\|$ and obtain some $L^p$-generalizations of the well-known Caffarelli contraction theorem. We establish a connection of our results with the Talagrand transportation inequality. We also prove a corresponding dimension-free version for the relative Fisher information with respect to a Gaussian measure.

math.PR↗