  In this paper we intend to give a comprehensive approach of functional
inequalities for diffusion processes under some "curvature" assumptions. Our
notion of curvature coincides with the usual $\Gamma_2$ curvature of Bakry and
Emery in the case of a (reversible) drifted Brownian motion, but differs for
more general diffusion processes. Our approach using simple coupling arguments
together with classical stochastic tools, allows us to obtain new results, to
recover and to extend already known results, giving in many situations explicit
(though non optimal) bounds. In particular, we show new results for
gradient/semigroup commutation in the log concave case. Some new convergence to
equilibrium in the granular media equation is also exhibited.
