Random walks in $(\mathbb {Z}_{+})^{2}$ with non-zero drift absorbed at the axes

Kurkova, Irina; Raschel, Kilian

Random walks in

{(ℤ_{+})}^{2}

with non-zero drift absorbed at the axes
[Marches aléatoires dans

ℤ_{+}^{2}

avec un drift non nul, absorbées au bord]

Kurkova, Irina ; Raschel, Kilian

Bulletin de la Société Mathématique de France, Tome 139 (2011), p. 341-387 / Harvested from Numdam

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Résumé
Abstract

Dans cet article, nous étudions les marches aléatoires du quart de plan ayant des sauts à distance au plus un, avec un drift non nul à l'intérieur et absorbées au bord. Nous obtenons de façon explicite les séries génératrices des probabilités d'absorption au bord, puis leur asymptotique lorsque le site d'absorption tend vers l'infini. Nous calculons également l'asymptotique des fonctions de Green le long de toutes les trajectoires, en particulier selon celles tangentes aux axes.

Spatially homogeneous random walks in ${(ℤ_{+})}^{2}$ with non-zero jump probabilities at distance at most $1$ , with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes.

Publié le : 2011-01-01

MR 2869310 | Zbl 1243.60042

DOI : https://doi.org/10.24033/bsmf.2611
Classification: 60G50, 60G40, 30E20, 30F10
Mots clés: marche aléatoire, fonctions de Green, probabilités d'absorption, singularités de fonctions complexes, prolongement analytique, méthode de la plus grande descente

@article{BSMF_2011__139_3_341_0,
     author = {Kurkova, Irina and Raschel, Kilian},
     title = {Random walks in $(\mathbb {Z}\_{+})^{2}$ with non-zero drift absorbed at the axes},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     volume = {139},
     year = {2011},
     pages = {341-387},
     doi = {10.24033/bsmf.2611},
     mrnumber = {2869310},
     zbl = {1243.60042},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BSMF_2011__139_3_341_0}
}

Kurkova, Irina; Raschel, Kilian. Random walks in $(\mathbb {Z}_{+})^{2}$ with non-zero drift absorbed at the axes. Bulletin de la Société Mathématique de France, Tome 139 (2011) pp. 341-387. doi : 10.24033/bsmf.2611. http://gdmltest.u-ga.fr/item/BSMF_2011__139_3_341_0/

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