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## Соболевская регулярность для бесконечномерного уравнения Монжа-Ампера

Доклады Академии наук. 2012. Т. 44. № 2. С. 131-136.

Колесников А. В., Богачев В. И.

Bogachev V. Providence: American Mathematical Society, 1998.

This book presents a systematic exposition of the modern theory of Gaussian measures. The basic properties of finite and infinite dimensional Gaussian distributions, including their linear and nonlinear transformations, are discussed. The book is intended for graduate students and researchers in probability theory, mathematical statistics, functional analysis, and mathematical physics. It contains a lot of examples and exercises. The bibliography contains 844 items; the detailed bibliographical comments and subject index are included.

Added: Mar 10, 2014

Zaev D. arxiv.org. math. Cornell University, 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 invariant measures and martingales. We formulate and prove a criterion for existence of a solution, a duality statement of the Kantorovich type, and a necessary geometric condition on a support of optimal measure, which is analogues to the usual c-monotonicity.

Added: May 14, 2014

Kolesnikov A. arxiv.org. math. Cornell University, 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 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 $M$.

Added: Mar 28, 2013

Bogachev V., Malofeev I. Potential analysis. 2016. Vol. 44. No. 4. P. 767-792.

We propose a new construction of surface measures on infinite-dimensional spaces

Added: Feb 1, 2017

Added: Mar 26, 2013

Bogachev V., Kolesnikov A. arxiv.org. math. Cornell University, 2011. No. 1110.1822..

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 $\varphi$.

Added: Mar 28, 2013

Added: Mar 23, 2011

Bogachev V. Providence: American Mathematical Society, 2010.

The book gives a systematic account of the theory of differentiable measures and the Malliavin calculus.

Added: Mar 5, 2014