  We use the framework used by Bakry and Emery in their work on logarithmic
Sobolev inequalities to define a notion of coarse Ricci curvature on smooth
metric measure spaces alternative to the notion proposed by Y. Ollivier. This
function can be used to recover the Ricci tensor on smooth Riemannian manifolds
by the formula $$ \mathrm{Ric}(\gamma^{\prime}\left( 0\right)
,\gamma^{\prime}\left( 0\right)
)=\frac{1}{2}\frac{d^{2}}{ds^{2}}\mathrm{Ric}_{\Delta_g}(x,\gamma\left(
s\right) )$$
  for any curve $\gamma(s).$
