  It is well-known that for a one dimensional stochastic differential equation
driven by Brownian noise, with coefficient functions satisfying the assumptions
of the Yamada-Watanabe theorem \cite{yamada1,yamada2} and the Feller test for
explosions \cite{feller51,feller54}, there exists a unique stationary
distribution with respect to the Markov semigroup of transition probabilities.
We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$
and study the rate of convergence to the stationary distribution. Using a
geometrical approach that uses the so called {\it free energy function} on the
density function space, we prove that the density functions, which are
solutions of the Fokker-Planck equation, converge to the stationary density
function exponentially under the Kullback-Leibler {divergence}, thus also in
the total variation norm. The results show that there is a relation between the
Bakry-Emery curvature dimension condition and the dissipativity condition of
the transformed system under the Fisher-Lamperti transformation. Several
applications are discussed, including the Cox-Ingersoll-Ross model and the
Ait-Sahalia model in finance and the Wright-Fisher model in population
genetics.
