Uniform convergence of conditional distributions for absorbed one-dimensional diffusions. (20th March 2018)

• Record Type: Journal Article
• Authors: Champagnat, Nicolas; Villemonais, Denis
• Is Part Of: Advances in applied probability. Volume 50:Number 1(2018)
• Page Start: 178
• Page End: 203
• Publication Date: 2018-03-20
• Subjects: One-dimensional diffusion, -- absorbed process, -- quasi-stationary distribution, -- uniform exponential mixing property, -- strict local martingale, -- one-dimensional process with jumps; Primary 60J60, -- 60J70, -- 37A25, -- 60B10, -- 60F99, -- Secondary 60G44, -- 60J75; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.9
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: In this paper we study the quasi-stationary behavior 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.