Optimal transportation and Ricci curvature for metric measure spaces.

We introduce and analyze generalized Ricci curvature bounds for
metric measure spaces $(M,d,m)$, based on convexity properties of
the relative entropy $Ent(. | m)$. For Riemannian manifolds,
$Curv(M,d,m)ge K$ if and only if $Ric_Mge K $ on $M$. For the
Wiener space, $Curv(M,d,m)=1$.

oindent One of the main results is that these lower curvature
bounds are stable under (e.g. measured Gromov-Hausdorff)
convergence.

medskip

oindent Moreover, we introduce a curvature-dimension condition
CD$(K,N)$ being more restrictive than the curvature bound
$Curv(M,d,m)ge K$.
For
Riemannian manifolds, CD$(K,N)$ is {equivalent} to $mbox{
m
Ric}_M(xi,xi)ge Kcdot |xi|^2$ and $mbox{
m dim}(M)le N$.

oindent Condition CD$(K,N)$ implies sharp version of the
Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison
theorem and of the Bonnet-Myers theorem. Moreover, it allows to
construct canonical Dirichlet forms with {Gaussian upper and lower
bounds} for the corresponding heat kernels.