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Regular version of the site

Article

On Sobolev Regularity of Mass Transport and Transportation Inequalities

Theory of Probability and Its Applications. 2013. Vol. 57. No. 2. P. 243-264.
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 ${\bf R}^d$. 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 addition, we prove a similar estimate for the $L^p(\mu)$-norms of $\|D^2 \Phi\|$ and obtain some $L^p$-generalizations of the well-known Caffarelli contraction theorem. We establish a connection between our results and the Talagrand transportation inequality. We also prove a corresponding dimension-free version for the relative Fisher information with respect to the Gaussian measure.