Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

TL;DR

This paper investigates the conditions under which absorbed one-dimensional diffusions converge exponentially to a unique quasi-stationary distribution uniformly across initial states, using tools like local martingale diffusions and providing various examples.

Contribution

It establishes necessary and sufficient conditions for uniform exponential convergence to quasi-stationarity in one-dimensional diffusions, including processes with jumps and sticky Brownian motion.

Findings

Exponential convergence to quasi-stationary distribution is characterized by specific conditions.

Expectation of local martingale diffusions is uniformly bounded over initial positions.

Examples include sticky Brownian motion and jump processes.

Abstract

This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one dimensional strict local martingale diffusions coming down from infinity. We prove under mild assumptions that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.