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It is well-known that for a one dimensional stochastic differential equationdriven by Brownian noise, with coefficient functions satisfying the assumptionsof the Yamada-Watanabe theorem \cite{yamada1,yamada2} and the Feller test forexplosions \cite{feller51,feller54}, there exists a unique stationarydistribution 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 ageometrical approach that uses the so called {\it free energy function} on thedensity function space, we prove that the density functions, which aresolutions of the Fokker-Planck equation, converge to the stationary densityfunction exponentially under the Kullback-Leibler {divergence}, thus also inthe total variation norm. The results show that there is a relation between theBakry-Emery curvature dimension condition and the dissipativity condition ofthe transformed system under the Fisher-Lamperti transformation. Severalapplications are discussed, including the Cox-Ingersoll-Ross model and theAit-Sahalia model in finance and the Wright-Fisher model in populationgenetics.

Ergodicity of scalar stochastic differential equations with H\"older continuous coefficients