Integrability of absolutely continuous transformations of measures and applications to optimal mass transportation. (English. Russian original) Zbl 1118.26015

Bogachev, V. I.; Kolesnikov, A. V.

Theory Probab. Appl. 50, No. 3, 367-385 (2006); translation from Teor. Veroyatn. Primen. 50, No. 3, 433-456 (2005).

The authors establish various extensions of Talagrand’s inequality of the following form. Let \(T(x) = x + F (x)\) be an optimal mass transportation from meausure \(\mu\) to measure \(\nu = g \cdot \mu\) on \(\mathbb{R}^d\), then \(\int \Phi_1 (| F(x)| ) \,d \mu (x) \leq \int \Phi_2 (g (x)) \,d \mu (x)\) for certain functions \(\Phi_1, \Phi_2\). Talagrand’s result is the case, where \(\Phi_1 (x) = x^2\) and \(\Phi_2 (x) = x\log x\) and \(\mu\) is a Gaussian measure on \(\mathbb{R}^d\). Examples included are, where \(\mu\) is Gaussian (or satisfies the logarithmic Sobolev inequality), \(\Phi_2 (x) = x^p\) and \(\Phi_1 (x) = \exp (\alpha x^2)\), or \(\Phi_2 (x) = x | \log x| ^p\) and \(\Phi_1 (x) = | x| ^r\). Some general inequalities of this type are established for \(\mu\) Gaussian or strictly convex measures and for optimal transformations w.r.t. cost functions \(| x-y| ^p\), \(p \in (1,2]\). For the case \(p=2\) also measures satisfying the logarithmic Sobolev inequality and the Poincaré inequality are considered and, e.g., \(L^p\)-estimates of the optimal transportation are given.

Reviewer: Ludger Rüschendorf (Freiburg i. Br.)

MSC:

26D10	Inequalities involving derivatives and differential and integral operators
60E15	Inequalities; stochastic orderings

Keywords:

optimal transportation; Gaussian measure; convex measure; logarithmic Sobolev inequality; Poincaré inequality; Talagrand inequality