Hitting time of a corner for a reflected diffusion in the square

F. Delarue

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 5, page 946-961
  • ISSN: 0246-0203

Abstract

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We discuss the long time behavior of a two-dimensional reflected diffusion in the unit square and investigate more specifically the hitting time of a neighborhood of the origin. We distinguish three different regimes depending on the sign of the correlation coefficient of the diffusion matrix at the point 0. For a positive correlation coefficient, the expectation of the hitting time is uniformly bounded as the neighborhood shrinks. For a negative one, the expectation explodes in a polynomial way as the diameter of the neighborhood vanishes. In the null case, the expectation explodes at a logarithmic rate. As a by-product, we establish in the different cases the attainability or nonattainability of the origin for the reflected process. From a practical point of view, the considered hitting time appears as a deadlock time in various resource sharing problems.

How to cite

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Delarue, F.. "Hitting time of a corner for a reflected diffusion in the square." Annales de l'I.H.P. Probabilités et statistiques 44.5 (2008): 946-961. <http://eudml.org/doc/77998>.

@article{Delarue2008,
abstract = {We discuss the long time behavior of a two-dimensional reflected diffusion in the unit square and investigate more specifically the hitting time of a neighborhood of the origin. We distinguish three different regimes depending on the sign of the correlation coefficient of the diffusion matrix at the point 0. For a positive correlation coefficient, the expectation of the hitting time is uniformly bounded as the neighborhood shrinks. For a negative one, the expectation explodes in a polynomial way as the diameter of the neighborhood vanishes. In the null case, the expectation explodes at a logarithmic rate. As a by-product, we establish in the different cases the attainability or nonattainability of the origin for the reflected process. From a practical point of view, the considered hitting time appears as a deadlock time in various resource sharing problems.},
author = {Delarue, F.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {reflected diffusions; hitting times; Lyapunov functions; distributed algorithms},
language = {eng},
number = {5},
pages = {946-961},
publisher = {Gauthier-Villars},
title = {Hitting time of a corner for a reflected diffusion in the square},
url = {http://eudml.org/doc/77998},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Delarue, F.
TI - Hitting time of a corner for a reflected diffusion in the square
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 5
SP - 946
EP - 961
AB - We discuss the long time behavior of a two-dimensional reflected diffusion in the unit square and investigate more specifically the hitting time of a neighborhood of the origin. We distinguish three different regimes depending on the sign of the correlation coefficient of the diffusion matrix at the point 0. For a positive correlation coefficient, the expectation of the hitting time is uniformly bounded as the neighborhood shrinks. For a negative one, the expectation explodes in a polynomial way as the diameter of the neighborhood vanishes. In the null case, the expectation explodes at a logarithmic rate. As a by-product, we establish in the different cases the attainability or nonattainability of the origin for the reflected process. From a practical point of view, the considered hitting time appears as a deadlock time in various resource sharing problems.
LA - eng
KW - reflected diffusions; hitting times; Lyapunov functions; distributed algorithms
UR - http://eudml.org/doc/77998
ER -

References

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