Колесников Александр Викторович

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

2012 · PREPRINT · en

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.

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

2012 · PREPRINT · en

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 metric-measure space $M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry-{\'E}mery tensor provided both $V$ and $W$ are convex. If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ 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 $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.

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

2012 · PREPRINT · en

Let $\gamma$ 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 $$ \dot{\rho} + \mbox{div}_{\gamma} (\rho \cdot {b}) =0, \ \ \rho|_{t=0} = \rho_0, $$ where $\rho_0 \cdot \gamma $ is a probability measure, admits a weak solution, in particular, under the following assumptions: $$ \|b\|_{H} \in L^p(\gamma), \ p>1, \ \ \ \exp\bigl(\varepsilon(\mbox{\rm div}_{\gamma} b)_{-} \bigr) \in L^1(\gamma). $$ Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures $\nu$ on $\R^{\infty}$, under the main assumption that $\beta_i \in \cap_{n \in \Nat} L^{n}(\nu)$ for every $i \in \Nat$, where $\beta_i$ is the logarithmic derivative of $\nu$ 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.

On isoperimetric sets of radially symmetric measures

2011 · ARTICLE · en

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. It is shown, 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.

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

2011 · PREPRINT · en

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

Транспортировка масс и сжимающие отображения

2010 · ARTICLE · ru

Согласно известному результату Л. Каффарелли, оптимальная транспортировка стандартной гауссовской меры в логарифмически вогнутую меру является 1-липшицевым отображением. Настоящая работа представляет собой краткий обзор различных результатов и приложений, полученных в этом направлении.

Глобальные гёльдеровы оценки для оптимальной транспортировки

2010 · ARTICLE · ru

О приложениях задачи транспортировки мер

2009 · ARTICLE · ru

Mass transport generated by a flow of Gauss maps

2009 · ARTICLE · en

Weak regularity of Gauss mass transport

2009 · PREPRINT · en

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 $\phi$, 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.