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Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

  • Nicolas Champagnat (a1) and Denis Villemonais (a2)
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.

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Copyright
Corresponding author
* 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.
** Email address: nicolas.champagnat@inria.fr
*** Email address: denis.villemonais@univ-lorraine.fr
References
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
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