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Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

Part of: Probability theory on algebraic and topological structures Ergodic theory Stochastic processes Limit theorems Markov processes

Published online by Cambridge University Press:  20 March 2018

Nicolas Champagnat  and
Denis Villemonais
Nicolas Champagnat*
Affiliation:
Inria Nancy Grand Est
Denis Villemonais*
Affiliation:
Université de Lorraine
*
* Postal address: Institut Élie Cartan de Lorraine (IECL, UMR CNRS 7502), Université de Lorraine, Campus Scientifique, B.P. 70239, Vandœuvre-lès-Nancy Cedex, F-54506, France.
* Postal address: Institut Élie Cartan de Lorraine (IECL, UMR CNRS 7502), Université de Lorraine, Campus Scientifique, B.P. 70239, Vandœuvre-lès-Nancy Cedex, F-54506, France.
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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.

Keywords

One-dimensional diffusionabsorbed processquasi-stationary distributionuniform exponential mixing propertystrict local martingaleone-dimensional process with jumps

MSC classification

Primary: 60J60: Diffusion processes 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 37A25: Ergodicity, mixing, rates of mixing 60B10: Convergence of probability measures 60F99: None of the above, but in this section
Secondary: 60G44: Martingales with continuous parameter 60J75: Jump processes
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 178 - 203
Copyright
Copyright © Applied Probability Trust 2018 

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