Hitting time of a corner for a reflected diffusion in the square
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 946-961.

Nous étudions le comportement en temps long d'une diffusion réfléchie à valeurs dans le carré unité et nous focalisons plus précisément sur le temps d'atteinte d'un voisinage de l'origine. Nous distinguons trois régimes différents, selon le signe du coefficient de corrélation de la matrice de diffusion prise au point 0. Pour un coefficient de corrélation strictement positif, l'espérance du temps d'atteinte reste bornée lorsque le voisinage se rétrécit. Pour un coefficient strictement négatif, l'espérance explose à vitesse polynomiale lorsque le diamètre du voisinage tend vers zéro. Dans le cas d'un coefficient nul, l'espérance diverge à vitesse logarithmique. Au passage, nous établissons selon les cas la possibilité ou l'impossibilité pour la diffusion réfléchie d'atteindre l'origine. D'un point de vue pratique, le temps d'atteinte considéré apparaît comme un instant de blocage dans différents problèmes de partage de ressource.

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.

DOI : 10.1214/07-AIHP128
Classification : 60H10, 60G40, 68W15
Mots-clés : reflected diffusions, hitting times, Lyapunov functions, distributed algorithms 
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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, Tome 44 (2008) no. 5, pp. 946-961. doi : 10.1214/07-AIHP128. http://archive.numdam.org/articles/10.1214/07-AIHP128/

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