Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

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.