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Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

Published online by Cambridge University Press:  20 March 2018

Nicolas Champagnat
Affiliation:
Inria Nancy Grand Est
Denis Villemonais
Affiliation:
Université de Lorraine

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.

Type
Original Article
Copyright
Copyright © Applied Probability Trust 2018 

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