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

We study the optimal transportation mapping $\nabla Φ: \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $μ= e^{-V} \ dx$ onto another probability measure $ν= e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 Φ$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\mathbb{R}^d, g, μ)$. We prove, in particular, that $M$ admits a non-negative Bakry--{É}mery tensor provided both $V$ and $W$ are convex. If the target measure $ν$ is the Lebesgue measure on a convex set $Ω$ and $μ$ 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 $\| D^2 Φ\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.