MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Ergodicity of scalar stochastic differential equations with Hoelder continuous coeffcients

Hoang Duc Luu, Tat Dat Tran and Jürgen Jost


It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coeffcient functions satisfying the assumptions of the Yamada-Watanabe theorem [30, 31] and the Feller test for explosions [17, 18], 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 R and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called 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.

stationary distributions, invariant measures, fokker-planck equation, kullback-leibler divergence, Cox-Ingersoll-Ross model, Ait-Sahalia model, Wright-Fisher model

Related publications

2018 Repository Open Access
Hoang Duc Luu, Tat Dat Tran and Jürgen Jost

Ergodicity of scalar stochastic differential equations with Hölder continuous coeffcients

In: Stochastic processes and their applications, 128 (2018) 10, pp. 3253-3272