Uniform convergence of conditional distributions for one-dimensional diffusion processes

In this paper, we study the quasi-stationary behavior of the one-dimensional diffusion process with a regular or exit boundary at 0 and an entrance boundary at $\infty$. By using the Doob's $h$-transform, we show that the conditional distribution of the process converges to its unique quasi-stationary distribution exponentially fast in the total variation norm, uniformly with respect to the initial distribution. Moreover, we also use the same method to show that the conditional distribution of the process converges exponentially fast in the $\psi$-norm to the unique quasi-stationary distribution. The rate of convergence of the conditional empirical measure to the quasi-ergodic distribution is also considered. Finally, two examples arising in population dynamics are also given to illustrate the main results.