Uniform convergence of conditional distributions for one-dimensional diffusion processes

TL;DR

This paper proves that the conditional distributions of certain one-dimensional diffusion processes converge exponentially fast to a unique quasi-stationary distribution, with uniform convergence rates in total variation and $\\psi$-norms, supported by examples from population dynamics.

Contribution

It establishes uniform exponential convergence of conditional distributions to the quasi-stationary distribution for one-dimensional diffusions using Doob's $h$-transform, extending understanding of their long-term behavior.

Findings

Conditional distributions converge exponentially fast in total variation norm.

Convergence also occurs in the $\psi$-norm with the same exponential rate.

Examples from population dynamics illustrate the theoretical results.

Abstract

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.