Hessian metrics and optimal transportation of log-concave measures

Details Hessian metrics and optimal transportation of log-concave measures Hessian metrics and optimal transportation of log-concave measures

2012-01-11

arxiv.org/abs/1201.2342v1

Mathematics

Analysis of PDEs

15 pages

Scientific paper

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 $(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $(\mathbb{R}^d, g, \mu)$ admits a non-negative Bakry-Emery tensor provided both $V$ and $W$ are convex. Applications of this result include some dimension-free bounds on $\| D^2 \Phi\|$ when $\mu$ is the standard Gaussian measure and $\nu$ is the Lebesgue measure on a convex set $K$. In addition, we obtain global Sobolev a priori estimates for measures with integrable logarithmic derivatives.

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