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 
@article{AIHPB_2008__44_5_946_0,
     author = {Delarue, F.},
     title = {Hitting time of a corner for a reflected diffusion in the square},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {946--961},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {5},
     year = {2008},
     doi = {10.1214/07-AIHP128},
     mrnumber = {2453777},
     zbl = {1180.60035},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/07-AIHP128/}
}
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
SP  - 946
EP  - 961
VL  - 44
IS  - 5
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/07-AIHP128/
DO  - 10.1214/07-AIHP128
LA  - en
ID  - AIHPB_2008__44_5_946_0
ER  - 
%0 Journal Article
%A Delarue, F.
%T Hitting time of a corner for a reflected diffusion in the square
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2008
%P 946-961
%V 44
%N 5
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/07-AIHP128/
%R 10.1214/07-AIHP128
%G en
%F AIHPB_2008__44_5_946_0
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://www.numdam.org/articles/10.1214/07-AIHP128/

[1] S. Balaji and S. Ramasubramanian. Passage time moments for multidimensional diffusions. J. Appl. Probab. 37 (2000) 246-251. | MR | Zbl

[2] F. Comets, F. Delarue and R. Schott. Distributed algorithms in an ergodic Markovian environment. Random Structures Algorithms 30 (2007) 131-167.

[3] J. G. Dai and R. J. Williams. Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedra. Theory Probab. Appl. 40 (1996) 1-40. | MR

[4] J. G. Dai and R. J. Williams. Letter to the editors: Remarks on our paper “existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedra”. Theory Probab. Appl. 50 (2006) 346-347. | MR | Zbl

[5] R. D. Deblassie. Explicit semimartingale representation of Brownian motion in a wedge. Stochastic Process. Appl. 34 (1990) 67-97. | MR | Zbl

[6] R. D. Deblassie, D. Hobson, E. A. Housworth and E. H. Toby. Escape rates for transient reflected Brownian motion in wedges and cones. Stochastics Stochastics Rep. 57 (1996) 199-211. | MR | Zbl

[7] G. Fayolle, V. A. Malyshev and M. V. Menshikov. Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press, Cambridge, 1995. | MR | Zbl

[8] A. Friedman. Stochastic Differential Equations and Applications. Vol. 1. Probability and Mathematical Statistics, Academic Press, New York-London, 1975. | MR | Zbl

[9] N. Guillotin-Plantard and R. Schott. Distributed algorithms with dynamic random transitions. Random Structures Algorithms 21 (2002) 371-396. | MR | Zbl

[10] R. Z. Hasminskii. Stochastic Stability of Differential Equations. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md, 1980. | MR | Zbl

[11] Y. Kwon and R. J. Williams. Reflected Brownian motion in a cone with radially homogeneous reflection field. Trans. Amer. Math. Soc. 327 (1991) 739-780. | MR | Zbl

[12] P.-L. Lions and A.-S. Sznitman. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511-537. | MR | Zbl

[13] G. Louchard. Some distributed algorithms revisited. Commun. Statist. Stochastic Models 11 (1995) 563-586. | MR | Zbl

[14] G. Louchard and R. Schott. Probabilistic analysis of some distributed algorithms. Random Structures Algorithms 2 (1991) 151-186. | MR | Zbl

[15] G. Louchard, R. Schott, M. Tolley and P. Zimmermann. Random walks, heat equations and distributed algorithms. J. Comput. Appl. Math. 53 (1994) 243-274. | MR | Zbl

[16] R. S. Maier. Colliding stacks: A large deviations analysis. Random Structures Algorithms 2 (1991) 379-420. | MR | Zbl

[17] R. S. Maier and R. Schott. Exhaustion of shared memory: stochastic results. In: Proceedings of WADS'93, LNCS No 709, Springer Verlag, 1993, pp. 494-505. | MR

[18] M. Menshikov and R. J. Williams. Passage-time moments for continuous non-negative stochastic processes and applications. Adv. in Appl. Probab. 28 (1996) 747-762. | MR | Zbl

[19] K. Ramanan. Reflected diffusions defined via the extended Skorokhod map. Electron. J. Probab. 11 (2006) 934-992. | MR | Zbl

[20] M. I. Reiman and R. J. Williams. A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77 (1988) 87-97. | MR | Zbl

[21] M. I. Reiman and R. J. Williams. Correction to: “A boundary property of semimartingale reflecting Brownian motions” [Probab. Theory Related Fields 77 87-97]. Probab. Theory Related Fields 80 (1989) 633. | MR | Zbl

[22] Y. Saisho. Stochastic differential equations for multidimensional domain with reflecting boundary. Probab. Theory Related Fields 74 (1987) 455-477. | MR | Zbl

[23] L. Słomiński. Euler's approximations of solutions of SDEs with reflecting boundary. Stochastic Process. Appl. 94 (2001) 317-337. | MR | Zbl

[24] H. Tanaka. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. 9 (1979) 163-177. | MR | Zbl

[25] L. M. Taylor and R. J. Williams. Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields 96 (1993) 283-317. | MR | Zbl

[26] S. R. S. Varadhan and R. J. Williams. Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 (1985) 405-443. | MR | Zbl

[27] R. J. Williams. Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Probab. 13 (1985) 758-778. | MR | Zbl

[28] R. J. Williams. Reflected Brownian motion in a wedge: semimartingale property. Z. Wahrsch. Verw. Gebiete 69 (1985) 161-176. | MR | Zbl

Cité par Sources :