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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Diffusions from infinity


Authors: Vincent Bansaye, Pierre Collet, Servet Martinez, Sylvie Méléard and Jaime San Martin
Journal: Trans. Amer. Math. Soc. 372 (2019), 5781-5823
MSC (2010): Primary 60J60, 60F05
DOI: https://doi.org/10.1090/tran/7841
Published electronically: June 19, 2019
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Abstract: In this paper we consider diffusions on the half line $ (0,\infty )$ such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity which takes finite values at positive times. We study the behavior of hitting times of large barriers and, in a dual way, the behavior of the process starting at infinity for small time. In particular, we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctuations of the hitting times are asymptotically Gaussian. We also derive the tail of the distribution of the hitting time of the origin and a Yaglom limit for the diffusion starting from infinity. We finally prove that the distribution of this process killed at the origin is absolutely continuous with respect to the speed measure. The density is expressed in terms of the eigenvalues and eigenfunctions of the generator of the killed diffusion.


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Additional Information

Vincent Bansaye
Affiliation: CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex, France
Email: vincent.bansaye@polytechnique.edu

Pierre Collet
Affiliation: CPHT, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex, France
Email: pierre.collet@cpht.polytechnique.fr

Servet Martinez
Affiliation: CMM-DIM, Universidad de Chile, UMI-CNRS 2807, Casilla 170-3, Correo 3, Santiago, Chile
Email: smartine@dim.uchile.cl

Sylvie Méléard
Affiliation: CMAP, Ecole Polytechnique, CNRS, route de Saclay, 91128 Palaiseau Cedex, France
Email: sylvie.meleard@polytechnique.edu

Jaime San Martin
Affiliation: CMM-DIM, Universidad de Chile, UMI-CNRS 2807, Casilla 170-3, Correo 3, Santiago, Chile
Email: jsanmart@dim.uchile.cl

DOI: https://doi.org/10.1090/tran/7841
Keywords: Diffusion, descent from infinity, entrance boundary, hitting times, eigenfunctions, quasi-stationary distributions, central limit theorem
Received by editor(s): January 29, 2018
Received by editor(s) in revised form: October 25, 2018
Published electronically: June 19, 2019
Additional Notes: This work was partially funded by the chair “Modélisation Mathématique et Biodiversité" of VEOLIA-Ecole Polytechnique-MnHn-FX, ANR-ABIM 16-CE40-0001 and by CMM Basal project AFB170001.
Article copyright: © Copyright 2019 American Mathematical Society